First, make the distinction between the common usage of "altruism" and "ethical altruism". Ethical altruism is essentially the doctrine that we exist for the sake of others. Comte coined the term and wrote:
The crucial point here is that justifying actions that benefit others on the basis that this will in turn benefit you is not proper in the context of ethical altruism. According to ethical altruism, you are obligated to serve others regardless. An argument that you ought to act in a way that benefits society for the reason that you will receive long term benefit is an inherently egoistic argument, not altruistic.
Be careful here. Proof presumes objective reality and the validity of our senses so there can be no "proof" that there is an objective reality or that our senses are valid. But it is the case that any attempt to "prove" that there is no objective reality or that our senses are not valid is stealing the concept of "proof" to deny the very concepts it depends on.
If I understand the alleged analytic/synthetic distinction correctly, the truth value of an "analytic" proposition depends only on the "meaning" of the concepts, i.e., "[the] predicate concept is contained in its subject concept". So, for example, the proposition "All bachelors are unmarried" is a so-called analytic proposition since "bachelor" is defined as an unmarried male. It is said that the negation of a true analytic proposition results in a contradiction, i.e., "all unmarried males are married" and so, true analytic propositions are necessarily true.
On the other hand, the truth value of the alleged "synthetic" proposition "All bachelors are less than 8 feet tall" does not depend on the "meaning" of the concepts, i.e., the definition of the concept "bachelor" does not involve height. Thus, it is said that a synthetic proposition, if true, is not necessarily true, i.e., the negation of a true synthetic proposition is not a contradiction.
According to those that accept the analytic/synthetic distinction, the "meaning" of a concept is the concept's definition.
However, according to Objectivism, "The meaning of a concept consists of the units—the existents—which it integrates, including all the characteristics of these units" and so the height of unmarried males is contained in the meaning of the concept "bachelor". The definition of a concept may change - it may become more refined - as new discoveries occur but the meaning of a concept doesn't change, the same units are referred to.
Now, consider the case that there is (or was or will be) a bachelor that is greater than 8 feet tall. According to Objectivism, the alleged synthetic proposition "All bachelors are less than 8 feet tall" becomes "All unmarried males including that one there that is greater than 8 feet tall are less than 8 feet tall" which is indeed a contradiction, i.e., it is a necessarily false proposition.
I'd like to write more but I'm pressed for time so this will have to do for now.