Theorem carddom

Description: Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of Quine p. 232. (Contributed by NM, 22-Oct-2003) (Revised by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Assertion	carddom	|- ( ( A e. V /\ B e. W ) -> ( ( card ` A ) C_ ( card ` B ) <-> A ~<_ B ) )

Proof

Step	Hyp	Ref	Expression
1		numth3	|- ( A e. V -> A e. dom card )
2		numth3	|- ( B e. W -> B e. dom card )
3		carddom2	|- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) <-> A ~<_ B ) )
4	1 2 3	syl2an	|- ( ( A e. V /\ B e. W ) -> ( ( card ` A ) C_ ( card ` B ) <-> A ~<_ B ) )