Metamath Proof Explorer


Theorem oncardid

Description: Any ordinal number is equinumerous to its cardinal number. Unlike cardid , this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004)

Ref Expression
Assertion oncardid
|- ( A e. On -> ( card ` A ) ~~ A )

Proof

Step Hyp Ref Expression
1 onenon
 |-  ( A e. On -> A e. dom card )
2 cardid2
 |-  ( A e. dom card -> ( card ` A ) ~~ A )
3 1 2 syl
 |-  ( A e. On -> ( card ` A ) ~~ A )