The "something resembling the pigeonhole principle" is the fact that if the union of finitely many sets is infinite, then at least one of the sets must be infinite as well.
I don't know whether it has a name.
I don't know whether it has a name.
Assume that 70 million disjoint sets all have finite cardinality. Then the cardinality of the union is the sum of the cardinalities, and is finite by construction.
Contradiction; QED.