Theorem carddom2 8271

Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8842, 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 8264	. 2 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) -> A ~<_ B ) )
2		brdom2 7464	. . 3 |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
3		cardon 8238	. . . . . . . 8 |- ( card ` A ) e. On
4	3	onelssi 4900	. . . . . . 7 |- ( ( card ` B ) e. ( card ` A ) -> ( card ` B ) C_ ( card ` A ) )
5		carddomi2 8264	. . . . . . . 8 |- ( ( B e. dom card /\ A e. dom card ) -> ( ( card ` B ) C_ ( card ` A ) -> B ~<_ A ) )
6	5	ancoms 451	. . . . . . 7 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` B ) C_ ( card ` A ) -> B ~<_ A ) )
7		domnsym 7562	. . . . . . 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 8238	. . . . . 6 |- ( card ` B ) e. On
11		ontri1 4826	. . . . . 6 |- ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) ) )
12	3, 10, 11	mp2an 670	. . . . 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 8261	. . . . . 6 |- ( A ~~ B -> ( card ` A ) = ( card ` B ) )
15		eqimss 3469	. . . . . 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 378	. . 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 366   /\ wa 367   = wceq 1399   e. wcel 1826   C_ wss 3389   class class class wbr 4367   Oncon0 4792   dom cdm 4913   ` cfv 5496   ~~ cen 7432   ~<_ cdom 7433   ~< csdm 7434   cardccrd 8229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-card 8233

This theorem is referenced by:  carduni  8275  carden2  8281  cardsdom2  8282  domtri2  8283  infxpidm2  8307  cardaleph  8383  infenaleph  8385  alephinit  8389  ficardun2  8496  ackbij2  8536  cfflb  8552  fin1a2lem9  8701  carddom  8842  pwfseqlem5  8952  hashdom  12350