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Metamath Proof Explorer

Theorem entric

Description: Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of Suppes p. 242. (Contributed by NM, 4-Jan-2004)

Ref Expression
Assertion entric 
|- ( ( A e. V /\ B e. W ) -> ( A ~< B \/ A ~~ B \/ B ~< A ) )

Proof

Step Hyp Ref Expression
1 domtri 
 |-  ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> -. B ~< A ) )
2 1 biimprd 
 |-  ( ( A e. V /\ B e. W ) -> ( -. B ~< A -> A ~<_ B ) )
3 brdom2 
 |-  ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
4 2 3 imbitrdi 
 |-  ( ( A e. V /\ B e. W ) -> ( -. B ~< A -> ( A ~< B \/ A ~~ B ) ) )
5 4 con1d 
 |-  ( ( A e. V /\ B e. W ) -> ( -. ( A ~< B \/ A ~~ B ) -> B ~< A ) )
6 5 orrd 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A ~< B \/ A ~~ B ) \/ B ~< A ) )
7 df-3or 
 |-  ( ( A ~< B \/ A ~~ B \/ B ~< A ) <-> ( ( A ~< B \/ A ~~ B ) \/ B ~< A ) )
8 6 7 sylibr 
 |-  ( ( A e. V /\ B e. W ) -> ( A ~< B \/ A ~~ B \/ B ~< A ) )