Theorem php2

Description: Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998) Avoid ax-pow . (Revised by BTernaryTau, 20-Nov-2024)

	Ref	Expression
Assertion	php2	|- ( ( A e. _om /\ B C. A ) -> B ~< A )

Proof

Step	Hyp	Ref	Expression
1		nnfi	|- ( A e. _om -> A e. Fin )
2		pssss	|- ( B C. A -> B C_ A )
3		ssdomfi	|- ( A e. Fin -> ( B C_ A -> B ~<_ A ) )
4	3	imp	|- ( ( A e. Fin /\ B C_ A ) -> B ~<_ A )
5	1 2 4	syl2an	|- ( ( A e. _om /\ B C. A ) -> B ~<_ A )
6		php	|- ( ( A e. _om /\ B C. A ) -> -. A ~~ B )
7		ensymfib	|- ( A e. Fin -> ( A ~~ B <-> B ~~ A ) )
8	7	biimprd	|- ( A e. Fin -> ( B ~~ A -> A ~~ B ) )
9	1 8	syl	|- ( A e. _om -> ( B ~~ A -> A ~~ B ) )
10	9	adantr	|- ( ( A e. _om /\ B C. A ) -> ( B ~~ A -> A ~~ B ) )
11	6 10	mtod	|- ( ( A e. _om /\ B C. A ) -> -. B ~~ A )
12		brsdom	|- ( B ~< A <-> ( B ~<_ A /\ -. B ~~ A ) )
13	5 11 12	sylanbrc	|- ( ( A e. _om /\ B C. A ) -> B ~< A )