Theorem carden2a 8792

Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 8793 are meant to replace carden 9373 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)

Assertion

Ref	Expression
carden2a	|- ( ( ( card ` A ) = ( card ` B ) /\ ( card ` A ) =/= (/) ) -> A ~~ B )

Proof of Theorem carden2a

Step	Hyp	Ref	Expression
1		df-ne 2795	. 2 |- ( ( card ` A ) =/= (/) <-> -. ( card ` A ) = (/) )
2		ndmfv 6218	. . . . . . 7 |- ( -. B e. dom card -> ( card ` B ) = (/) )
3		eqeq1 2626	. . . . . . 7 |- ( ( card ` A ) = ( card ` B ) -> ( ( card ` A ) = (/) <-> ( card ` B ) = (/) ) )
4	2, 3	syl5ibr 236	. . . . . 6 |- ( ( card ` A ) = ( card ` B ) -> ( -. B e. dom card -> ( card ` A ) = (/) ) )
5	4	con1d 139	. . . . 5 |- ( ( card ` A ) = ( card ` B ) -> ( -. ( card ` A ) = (/) -> B e. dom card ) )
6	5	imp 445	. . . 4 |- ( ( ( card ` A ) = ( card ` B ) /\ -. ( card ` A ) = (/) ) -> B e. dom card )
7		cardid2 8779	. . . 4 |- ( B e. dom card -> ( card ` B ) ~~ B )
8	6, 7	syl 17	. . 3 |- ( ( ( card ` A ) = ( card ` B ) /\ -. ( card ` A ) = (/) ) -> ( card ` B ) ~~ B )
9		cardid2 8779	. . . . . . 7 |- ( A e. dom card -> ( card ` A ) ~~ A )
10		ndmfv 6218	. . . . . . 7 |- ( -. A e. dom card -> ( card ` A ) = (/) )
11	9, 10	nsyl4 156	. . . . . 6 |- ( -. ( card ` A ) = (/) -> ( card ` A ) ~~ A )
12	11	ensymd 8007	. . . . 5 |- ( -. ( card ` A ) = (/) -> A ~~ ( card ` A ) )
13		breq2 4657	. . . . . 6 |- ( ( card ` A ) = ( card ` B ) -> ( A ~~ ( card ` A ) <-> A ~~ ( card ` B ) ) )
14		entr 8008	. . . . . . 7 |- ( ( A ~~ ( card ` B ) /\ ( card ` B ) ~~ B ) -> A ~~ B )
15	14	ex 450	. . . . . 6 |- ( A ~~ ( card ` B ) -> ( ( card ` B ) ~~ B -> A ~~ B ) )
16	13, 15	syl6bi 243	. . . . 5 |- ( ( card ` A ) = ( card ` B ) -> ( A ~~ ( card ` A ) -> ( ( card ` B ) ~~ B -> A ~~ B ) ) )
17	12, 16	syl5 34	. . . 4 |- ( ( card ` A ) = ( card ` B ) -> ( -. ( card ` A ) = (/) -> ( ( card ` B ) ~~ B -> A ~~ B ) ) )
18	17	imp 445	. . 3 |- ( ( ( card ` A ) = ( card ` B ) /\ -. ( card ` A ) = (/) ) -> ( ( card ` B ) ~~ B -> A ~~ B ) )
19	8, 18	mpd 15	. 2 |- ( ( ( card ` A ) = ( card ` B ) /\ -. ( card ` A ) = (/) ) -> A ~~ B )
20	1, 19	sylan2b 492	1 |- ( ( ( card ` A ) = ( card ` B ) /\ ( card ` A ) =/= (/) ) -> A ~~ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ` cfv 5888   ~~ cen 7952   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-card 8765

This theorem is referenced by:  card1  8794