Theorem ackardcard

Description: The Axiom of Choice implies that two sets have equal kard cardinalities iff they have equal card cardinalities. (Contributed by BTernaryTau, 3-Jul-2026)

	Ref	Expression
Assertion	ackardcard	Could not format assertion : No typesetting found for |- ( CHOICE -> ( ( A e. V /\ B e. W ) -> ( ( kard ` A ) = ( kard ` B ) <-> ( card ` A ) = ( card ` B ) ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		acnum	$$⊢\mathrm{CHOICE}\to \left(A\in V\to A\in \mathrm{dom}\mathrm{card}\right)$$
2		acnum	$$⊢\mathrm{CHOICE}\to \left(B\in W\to B\in \mathrm{dom}\mathrm{card}\right)$$
3	1 2	anim12d	$$⊢\mathrm{CHOICE}\to \left(\left(A\in V\wedge B\in W\right)\to \left(A\in \mathrm{dom}\mathrm{card}\wedge B\in \mathrm{dom}\mathrm{card}\right)\right)$$
4		kardcard2	Could not format ( ( A e. dom card /\ B e. dom card ) -> ( ( kard ` A ) = ( kard ` B ) <-> ( card ` A ) = ( card ` B ) ) ) : No typesetting found for |- ( ( A e. dom card /\ B e. dom card ) -> ( ( kard ` A ) = ( kard ` B ) <-> ( card ` A ) = ( card ` B ) ) ) with typecode |-
5	3 4	syl6	Could not format ( CHOICE -> ( ( A e. V /\ B e. W ) -> ( ( kard ` A ) = ( kard ` B ) <-> ( card ` A ) = ( card ` B ) ) ) ) : No typesetting found for |- ( CHOICE -> ( ( A e. V /\ B e. W ) -> ( ( kard ` A ) = ( kard ` B ) <-> ( card ` A ) = ( card ` B ) ) ) ) with typecode |-