Theorem carddom2 8139

Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8710, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
carddom2	|- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) <-> A ~<_ B ) )

Proof of Theorem carddom2

Step	Hyp	Ref	Expression
1		carddomi2 8132	. 2 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) -> A ~<_ B ) )
2		brdom2 7331	. . 3 |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
3		cardon 8106	. . . . . . . 8 |- ( card ` A ) e. On
4	3	onelssi 4822	. . . . . . 7 |- ( ( card ` B ) e. ( card ` A ) -> ( card ` B ) C_ ( card ` A ) )
5		carddomi2 8132	. . . . . . . 8 |- ( ( B e. dom card /\ A e. dom card ) -> ( ( card ` B ) C_ ( card ` A ) -> B ~<_ A ) )
6	5	ancoms 453	. . . . . . 7 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` B ) C_ ( card ` A ) -> B ~<_ A ) )
7		domnsym 7429	. . . . . . 7 |- ( B ~<_ A -> -. A ~< B )
8	4, 6, 7	syl56 34	. . . . . 6 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` B ) e. ( card ` A ) -> -. A ~< B ) )
9	8	con2d 115	. . . . 5 |- ( ( A e. dom card /\ B e. dom card ) -> ( A ~< B -> -. ( card ` B ) e. ( card ` A ) ) )
10		cardon 8106	. . . . . 6 |- ( card ` B ) e. On
11		ontri1 4748	. . . . . 6 |- ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) ) )
12	3, 10, 11	mp2an 672	. . . . 5 |- ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) )
13	9, 12	syl6ibr 227	. . . 4 |- ( ( A e. dom card /\ B e. dom card ) -> ( A ~< B -> ( card ` A ) C_ ( card ` B ) ) )
14		carden2b 8129	. . . . . 6 |- ( A ~~ B -> ( card ` A ) = ( card ` B ) )
15		eqimss 3403	. . . . . 6 |- ( ( card ` A ) = ( card ` B ) -> ( card ` A ) C_ ( card ` B ) )
16	14, 15	syl 16	. . . . 5 |- ( A ~~ B -> ( card ` A ) C_ ( card ` B ) )
17	16	a1i 11	. . . 4 |- ( ( A e. dom card /\ B e. dom card ) -> ( A ~~ B -> ( card ` A ) C_ ( card ` B ) ) )
18	13, 17	jaod 380	. . 3 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( A ~< B \/ A ~~ B ) -> ( card ` A ) C_ ( card ` B ) ) )
19	2, 18	syl5bi 217	. 2 |- ( ( A e. dom card /\ B e. dom card ) -> ( A ~<_ B -> ( card ` A ) C_ ( card ` B ) ) )
20	1, 19	impbid 191	1 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) <-> A ~<_ B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   C_ wss 3323   class class class wbr 4287   Oncon0 4714   dom cdm 4835   ` cfv 5413   ~~ cen 7299   ~<_ cdom 7300   ~< csdm 7301   cardccrd 8097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-card 8101

This theorem is referenced by:  carduni  8143  carden2  8149  cardsdom2  8150  domtri2  8151  infxpidm2  8175  cardaleph  8251  infenaleph  8253  alephinit  8257  ficardun2  8364  ackbij2  8404  cfflb  8420  fin1a2lem9  8569  carddom  8710  pwfseqlem5  8822  hashdom  12134