Theorem kardcard2a

Description: If two sets have equal nonzero card cardinalities, then they have equal kard cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026)

	Ref	Expression
Assertion	kardcard2a	Could not format assertion : No typesetting found for |- ( ( ( card ` A ) = ( card ` B ) /\ ( card ` A ) =/= (/) ) -> ( kard ` A ) = ( kard ` B ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		carden2a	$$⊢\left(\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\wedge \mathrm{card}\left(A\right)\ne \varnothing \right)\to A\approx B$$
2		fvfundmfvn0	$$⊢\mathrm{card}\left(A\right)\ne \varnothing \to \left(A\in \mathrm{dom}\mathrm{card}\wedge \mathrm{Fun}\left({\mathrm{card}\upharpoonright }_{\left\{A\right\}}\right)\right)$$
3	2	simpld	$$⊢\mathrm{card}\left(A\right)\ne \varnothing \to A\in \mathrm{dom}\mathrm{card}$$
4		kardeng	Could not format ( A e. dom card -> ( ( kard ` A ) = ( kard ` B ) <-> A ~~ B ) ) : No typesetting found for |- ( A e. dom card -> ( ( kard ` A ) = ( kard ` B ) <-> A ~~ B ) ) with typecode |-
5	3 4	syl	Could not format ( ( card ` A ) =/= (/) -> ( ( kard ` A ) = ( kard ` B ) <-> A ~~ B ) ) : No typesetting found for |- ( ( card ` A ) =/= (/) -> ( ( kard ` A ) = ( kard ` B ) <-> A ~~ B ) ) with typecode |-
6	5	adantl	Could not format ( ( ( card ` A ) = ( card ` B ) /\ ( card ` A ) =/= (/) ) -> ( ( kard ` A ) = ( kard ` B ) <-> A ~~ B ) ) : No typesetting found for |- ( ( ( card ` A ) = ( card ` B ) /\ ( card ` A ) =/= (/) ) -> ( ( kard ` A ) = ( kard ` B ) <-> A ~~ B ) ) with typecode |-
7	1 6	mpbird	Could not format ( ( ( card ` A ) = ( card ` B ) /\ ( card ` A ) =/= (/) ) -> ( kard ` A ) = ( kard ` B ) ) : No typesetting found for |- ( ( ( card ` A ) = ( card ` B ) /\ ( card ` A ) =/= (/) ) -> ( kard ` A ) = ( kard ` B ) ) with typecode |-