Theorem carddom2 8408

Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8976, 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 8401	. 2 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) -> A ~<_ B ) )
2		brdom2 7596	. . 3 |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
3		cardon 8375	. . . . . . . 8 |- ( card ` A ) e. On
4	3	onelssi 5530	. . . . . . 7 |- ( ( card ` B ) e. ( card ` A ) -> ( card ` B ) C_ ( card ` A ) )
5		carddomi2 8401	. . . . . . . 8 |- ( ( B e. dom card /\ A e. dom card ) -> ( ( card ` B ) C_ ( card ` A ) -> B ~<_ A ) )
6	5	ancoms 455	. . . . . . 7 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` B ) C_ ( card ` A ) -> B ~<_ A ) )
7		domnsym 7695	. . . . . . 7 |- ( B ~<_ A -> -. A ~< B )
8	4, 6, 7	syl56 35	. . . . . 6 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` B ) e. ( card ` A ) -> -. A ~< B ) )
9	8	con2d 119	. . . . 5 |- ( ( A e. dom card /\ B e. dom card ) -> ( A ~< B -> -. ( card ` B ) e. ( card ` A ) ) )
10		cardon 8375	. . . . . 6 |- ( card ` B ) e. On
11		ontri1 5456	. . . . . 6 |- ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) ) )
12	3, 10, 11	mp2an 677	. . . . 5 |- ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) )
13	9, 12	syl6ibr 231	. . . 4 |- ( ( A e. dom card /\ B e. dom card ) -> ( A ~< B -> ( card ` A ) C_ ( card ` B ) ) )
14		carden2b 8398	. . . . . 6 |- ( A ~~ B -> ( card ` A ) = ( card ` B ) )
15		eqimss 3483	. . . . . 6 |- ( ( card ` A ) = ( card ` B ) -> ( card ` A ) C_ ( card ` B ) )
16	14, 15	syl 17	. . . . 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 382	. . 3 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( A ~< B \/ A ~~ B ) -> ( card ` A ) C_ ( card ` B ) ) )
19	2, 18	syl5bi 221	. 2 |- ( ( A e. dom card /\ B e. dom card ) -> ( A ~<_ B -> ( card ` A ) C_ ( card ` B ) ) )
20	1, 19	impbid 194	1 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) <-> A ~<_ B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1443   e. wcel 1886   C_ wss 3403   class class class wbr 4401   dom cdm 4833   Oncon0 5422   ` cfv 5581   ~~ cen 7563   ~<_ cdom 7564   ~< csdm 7565   cardccrd 8366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-ord 5425  df-on 5426  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-card 8370

This theorem is referenced by:  carduni  8412  carden2  8418  cardsdom2  8419  domtri2  8420  infxpidm2  8445  cardaleph  8517  infenaleph  8519  alephinit  8523  ficardun2  8630  ackbij2  8670  cfflb  8686  fin1a2lem9  8835  carddom  8976  pwfseqlem5  9085  hashdom  12555