Theorem carden2b 8793

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8792 are meant to replace carden 9373 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	|- ( A ~~ B -> ( card ` A ) = ( card ` B ) )

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 8791	. . . . 5 |- ( ( card ` B ) e. ( card ` A ) -> -. ( card ` B ) ~~ A )
2		ennum 8773	. . . . . . . 8 |- ( A ~~ B -> ( A e. dom card <-> B e. dom card ) )
3	2	biimpa 501	. . . . . . 7 |- ( ( A ~~ B /\ A e. dom card ) -> B e. dom card )
4		cardid2 8779	. . . . . . 7 |- ( B e. dom card -> ( card ` B ) ~~ B )
5	3, 4	syl 17	. . . . . 6 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) ~~ B )
6		ensym 8005	. . . . . . 7 |- ( A ~~ B -> B ~~ A )
7	6	adantr 481	. . . . . 6 |- ( ( A ~~ B /\ A e. dom card ) -> B ~~ A )
8		entr 8008	. . . . . 6 |- ( ( ( card ` B ) ~~ B /\ B ~~ A ) -> ( card ` B ) ~~ A )
9	5, 7, 8	syl2anc 693	. . . . 5 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) ~~ A )
10	1, 9	nsyl3 133	. . . 4 |- ( ( A ~~ B /\ A e. dom card ) -> -. ( card ` B ) e. ( card ` A ) )
11		cardon 8770	. . . . 5 |- ( card ` A ) e. On
12		cardon 8770	. . . . 5 |- ( card ` B ) e. On
13		ontri1 5757	. . . . 5 |- ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) ) )
14	11, 12, 13	mp2an 708	. . . 4 |- ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) )
15	10, 14	sylibr 224	. . 3 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) C_ ( card ` B ) )
16		cardne 8791	. . . . 5 |- ( ( card ` A ) e. ( card ` B ) -> -. ( card ` A ) ~~ B )
17		cardid2 8779	. . . . . 6 |- ( A e. dom card -> ( card ` A ) ~~ A )
18		id 22	. . . . . 6 |- ( A ~~ B -> A ~~ B )
19		entr 8008	. . . . . 6 |- ( ( ( card ` A ) ~~ A /\ A ~~ B ) -> ( card ` A ) ~~ B )
20	17, 18, 19	syl2anr 495	. . . . 5 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) ~~ B )
21	16, 20	nsyl3 133	. . . 4 |- ( ( A ~~ B /\ A e. dom card ) -> -. ( card ` A ) e. ( card ` B ) )
22		ontri1 5757	. . . . 5 |- ( ( ( card ` B ) e. On /\ ( card ` A ) e. On ) -> ( ( card ` B ) C_ ( card ` A ) <-> -. ( card ` A ) e. ( card ` B ) ) )
23	12, 11, 22	mp2an 708	. . . 4 |- ( ( card ` B ) C_ ( card ` A ) <-> -. ( card ` A ) e. ( card ` B ) )
24	21, 23	sylibr 224	. . 3 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) C_ ( card ` A ) )
25	15, 24	eqssd 3620	. 2 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) = ( card ` B ) )
26		ndmfv 6218	. . . 4 |- ( -. A e. dom card -> ( card ` A ) = (/) )
27	26	adantl 482	. . 3 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` A ) = (/) )
28	2	notbid 308	. . . . 5 |- ( A ~~ B -> ( -. A e. dom card <-> -. B e. dom card ) )
29	28	biimpa 501	. . . 4 |- ( ( A ~~ B /\ -. A e. dom card ) -> -. B e. dom card )
30		ndmfv 6218	. . . 4 |- ( -. B e. dom card -> ( card ` B ) = (/) )
31	29, 30	syl 17	. . 3 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` B ) = (/) )
32	27, 31	eqtr4d 2659	. 2 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` A ) = ( card ` B ) )
33	25, 32	pm2.61dan 832	1 |- ( A ~~ B -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   (/)c0 3915   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` 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  carddom2  8803  cardennn  8809  cardsucinf  8810  pm54.43lem  8825  nnacda  9023  ficardun  9024  ackbij1lem5  9046  ackbij1lem8  9049  ackbij1lem9  9050  ackbij2lem2  9062  carden  9373  r1tskina  9604  cardfz  12769