Theorem carddomi2 8796

Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9376, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
carddomi2	|- ( ( A e. dom card /\ B e. V ) -> ( ( card ` A ) C_ ( card ` B ) -> A ~<_ B ) )

Proof of Theorem carddomi2

Step	Hyp	Ref	Expression
1		cardnueq0 8790	. . . . . 6 |- ( A e. dom card -> ( ( card ` A ) = (/) <-> A = (/) ) )
2	1	adantr 481	. . . . 5 |- ( ( A e. dom card /\ B e. V ) -> ( ( card ` A ) = (/) <-> A = (/) ) )
3	2	biimpa 501	. . . 4 |- ( ( ( A e. dom card /\ B e. V ) /\ ( card ` A ) = (/) ) -> A = (/) )
4		0domg 8087	. . . . 5 |- ( B e. V -> (/) ~<_ B )
5	4	ad2antlr 763	. . . 4 |- ( ( ( A e. dom card /\ B e. V ) /\ ( card ` A ) = (/) ) -> (/) ~<_ B )
6	3, 5	eqbrtrd 4675	. . 3 |- ( ( ( A e. dom card /\ B e. V ) /\ ( card ` A ) = (/) ) -> A ~<_ B )
7	6	a1d 25	. 2 |- ( ( ( A e. dom card /\ B e. V ) /\ ( card ` A ) = (/) ) -> ( ( card ` A ) C_ ( card ` B ) -> A ~<_ B ) )
8		fvex 6201	. . . . 5 |- ( card ` B ) e. _V
9		simprr 796	. . . . 5 |- ( ( ( A e. dom card /\ B e. V ) /\ ( ( card ` A ) =/= (/) /\ ( card ` A ) C_ ( card ` B ) ) ) -> ( card ` A ) C_ ( card ` B ) )
10		ssdomg 8001	. . . . 5 |- ( ( card ` B ) e. _V -> ( ( card ` A ) C_ ( card ` B ) -> ( card ` A ) ~<_ ( card ` B ) ) )
11	8, 9, 10	mpsyl 68	. . . 4 |- ( ( ( A e. dom card /\ B e. V ) /\ ( ( card ` A ) =/= (/) /\ ( card ` A ) C_ ( card ` B ) ) ) -> ( card ` A ) ~<_ ( card ` B ) )
12		cardid2 8779	. . . . . 6 |- ( A e. dom card -> ( card ` A ) ~~ A )
13	12	ad2antrr 762	. . . . 5 |- ( ( ( A e. dom card /\ B e. V ) /\ ( ( card ` A ) =/= (/) /\ ( card ` A ) C_ ( card ` B ) ) ) -> ( card ` A ) ~~ A )
14		simprl 794	. . . . . . 7 |- ( ( ( A e. dom card /\ B e. V ) /\ ( ( card ` A ) =/= (/) /\ ( card ` A ) C_ ( card ` B ) ) ) -> ( card ` A ) =/= (/) )
15		ssn0 3976	. . . . . . 7 |- ( ( ( card ` A ) C_ ( card ` B ) /\ ( card ` A ) =/= (/) ) -> ( card ` B ) =/= (/) )
16	9, 14, 15	syl2anc 693	. . . . . 6 |- ( ( ( A e. dom card /\ B e. V ) /\ ( ( card ` A ) =/= (/) /\ ( card ` A ) C_ ( card ` B ) ) ) -> ( card ` B ) =/= (/) )
17		ndmfv 6218	. . . . . . 7 |- ( -. B e. dom card -> ( card ` B ) = (/) )
18	17	necon1ai 2821	. . . . . 6 |- ( ( card ` B ) =/= (/) -> B e. dom card )
19		cardid2 8779	. . . . . 6 |- ( B e. dom card -> ( card ` B ) ~~ B )
20	16, 18, 19	3syl 18	. . . . 5 |- ( ( ( A e. dom card /\ B e. V ) /\ ( ( card ` A ) =/= (/) /\ ( card ` A ) C_ ( card ` B ) ) ) -> ( card ` B ) ~~ B )
21		domen1 8102	. . . . . 6 |- ( ( card ` A ) ~~ A -> ( ( card ` A ) ~<_ ( card ` B ) <-> A ~<_ ( card ` B ) ) )
22		domen2 8103	. . . . . 6 |- ( ( card ` B ) ~~ B -> ( A ~<_ ( card ` B ) <-> A ~<_ B ) )
23	21, 22	sylan9bb 736	. . . . 5 |- ( ( ( card ` A ) ~~ A /\ ( card ` B ) ~~ B ) -> ( ( card ` A ) ~<_ ( card ` B ) <-> A ~<_ B ) )
24	13, 20, 23	syl2anc 693	. . . 4 |- ( ( ( A e. dom card /\ B e. V ) /\ ( ( card ` A ) =/= (/) /\ ( card ` A ) C_ ( card ` B ) ) ) -> ( ( card ` A ) ~<_ ( card ` B ) <-> A ~<_ B ) )
25	11, 24	mpbid 222	. . 3 |- ( ( ( A e. dom card /\ B e. V ) /\ ( ( card ` A ) =/= (/) /\ ( card ` A ) C_ ( card ` B ) ) ) -> A ~<_ B )
26	25	expr 643	. 2 |- ( ( ( A e. dom card /\ B e. V ) /\ ( card ` A ) =/= (/) ) -> ( ( card ` A ) C_ ( card ` B ) -> A ~<_ B ) )
27	7, 26	pm2.61dane 2881	1 |- ( ( A e. dom card /\ B e. V ) -> ( ( card ` A ) C_ ( card ` B ) -> A ~<_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   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-dom 7957  df-card 8765

This theorem is referenced by:  carddom2  8803