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Metamath Proof Explorer

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 )