Theorem carden2a

Description: If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b are meant to replace carden in ZF without AC. (Contributed by Mario Carneiro, 9-Jan-2013)

	Ref	Expression
Assertion	carden2a	$$⊢\left(\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\wedge \mathrm{card}\left(A\right)\ne \varnothing \right)\to A\approx B$$

Proof

Step	Hyp	Ref	Expression
1		df-ne	$$⊢\mathrm{card}\left(A\right)\ne \varnothing \leftrightarrow \neg \mathrm{card}\left(A\right)=\varnothing$$
2		ndmfv	$$⊢\neg B\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(B\right)=\varnothing$$
3		eqeq1	$$⊢\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\to \left(\mathrm{card}\left(A\right)=\varnothing \leftrightarrow \mathrm{card}\left(B\right)=\varnothing \right)$$
4	2 3	imbitrrid	$$⊢\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\to \left(\neg B\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(A\right)=\varnothing \right)$$
5	4	con1d	$$⊢\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\to \left(\neg \mathrm{card}\left(A\right)=\varnothing \to B\in \mathrm{dom}\mathrm{card}\right)$$
6	5	imp	$$⊢\left(\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\wedge \neg \mathrm{card}\left(A\right)=\varnothing \right)\to B\in \mathrm{dom}\mathrm{card}$$
7		cardid2	$$⊢B\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(B\right)\approx B$$
8	6 7	syl	$$⊢\left(\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\wedge \neg \mathrm{card}\left(A\right)=\varnothing \right)\to \mathrm{card}\left(B\right)\approx B$$
9		breq2	$$⊢\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\to \left(A\approx \mathrm{card}\left(A\right)\leftrightarrow A\approx \mathrm{card}\left(B\right)\right)$$
10		entr	$$⊢\left(A\approx \mathrm{card}\left(B\right)\wedge \mathrm{card}\left(B\right)\approx B\right)\to A\approx B$$
11	10	ex	$$⊢A\approx \mathrm{card}\left(B\right)\to \left(\mathrm{card}\left(B\right)\approx B\to A\approx B\right)$$
12	9 11	biimtrdi	$$⊢\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\to \left(A\approx \mathrm{card}\left(A\right)\to \left(\mathrm{card}\left(B\right)\approx B\to A\approx B\right)\right)$$
13		cardid2	$$⊢A\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(A\right)\approx A$$
14		ndmfv	$$⊢\neg A\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(A\right)=\varnothing$$
15	13 14	nsyl4	$$⊢\neg \mathrm{card}\left(A\right)=\varnothing \to \mathrm{card}\left(A\right)\approx A$$
16	15	ensymd	$$⊢\neg \mathrm{card}\left(A\right)=\varnothing \to A\approx \mathrm{card}\left(A\right)$$
17	12 16	impel	$$⊢\left(\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\wedge \neg \mathrm{card}\left(A\right)=\varnothing \right)\to \left(\mathrm{card}\left(B\right)\approx B\to A\approx B\right)$$
18	8 17	mpd	$$⊢\left(\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\wedge \neg \mathrm{card}\left(A\right)=\varnothing \right)\to A\approx B$$
19	1 18	sylan2b	$$⊢\left(\mathrm{card}\left(A\right)=\mathrm{card}\left(B\right)\wedge \mathrm{card}\left(A\right)\ne \varnothing \right)\to A\approx B$$