Theorem carddom2 8354

Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8925, 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 8347	. 2 |- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) -> A ~<_ B ) )
2		brdom2 7542	. . 3 |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
3		cardon 8321	. . . . . . . 8 |- ( card ` A ) e. On
4	3	onelssi 4986	. . . . . . 7 |- ( ( card ` B ) e. ( card ` A ) -> ( card ` B ) C_ ( card ` A ) )
5		carddomi2 8347	. . . . . . . 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 7640	. . . . . . 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 8321	. . . . . 6 |- ( card ` B ) e. On
11		ontri1 4912	. . . . . 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 8344	. . . . . 6 |- ( A ~~ B -> ( card ` A ) = ( card ` B ) )
15		eqimss 3556	. . . . . 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 1379   e. wcel 1767   C_ wss 3476   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5586   ~~ cen 7510   ~<_ cdom 7511   ~< csdm 7512   cardccrd 8312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-card 8316

This theorem is referenced by:  carduni  8358  carden2  8364  cardsdom2  8365  domtri2  8366  infxpidm2  8390  cardaleph  8466  infenaleph  8468  alephinit  8472  ficardun2  8579  ackbij2  8619  cfflb  8635  fin1a2lem9  8784  carddom  8925  pwfseqlem5  9037  hashdom  12409