Theorem cardsdomel 8800

Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)

Assertion

Ref	Expression
cardsdomel	|- ( ( A e. On /\ B e. dom card ) -> ( A ~< B <-> A e. ( card ` B ) ) )

Proof of Theorem cardsdomel

Step	Hyp	Ref	Expression
1		cardid2 8779	. . . . . . 7 |- ( B e. dom card -> ( card ` B ) ~~ B )
2	1	ensymd 8007	. . . . . 6 |- ( B e. dom card -> B ~~ ( card ` B ) )
3		sdomentr 8094	. . . . . 6 |- ( ( A ~< B /\ B ~~ ( card ` B ) ) -> A ~< ( card ` B ) )
4	2, 3	sylan2 491	. . . . 5 |- ( ( A ~< B /\ B e. dom card ) -> A ~< ( card ` B ) )
5		ssdomg 8001	. . . . . . . 8 |- ( A e. On -> ( ( card ` B ) C_ A -> ( card ` B ) ~<_ A ) )
6		cardon 8770	. . . . . . . . 9 |- ( card ` B ) e. On
7		domtriord 8106	. . . . . . . . 9 |- ( ( ( card ` B ) e. On /\ A e. On ) -> ( ( card ` B ) ~<_ A <-> -. A ~< ( card ` B ) ) )
8	6, 7	mpan 706	. . . . . . . 8 |- ( A e. On -> ( ( card ` B ) ~<_ A <-> -. A ~< ( card ` B ) ) )
9	5, 8	sylibd 229	. . . . . . 7 |- ( A e. On -> ( ( card ` B ) C_ A -> -. A ~< ( card ` B ) ) )
10	9	con2d 129	. . . . . 6 |- ( A e. On -> ( A ~< ( card ` B ) -> -. ( card ` B ) C_ A ) )
11		ontri1 5757	. . . . . . . 8 |- ( ( ( card ` B ) e. On /\ A e. On ) -> ( ( card ` B ) C_ A <-> -. A e. ( card ` B ) ) )
12	6, 11	mpan 706	. . . . . . 7 |- ( A e. On -> ( ( card ` B ) C_ A <-> -. A e. ( card ` B ) ) )
13	12	con2bid 344	. . . . . 6 |- ( A e. On -> ( A e. ( card ` B ) <-> -. ( card ` B ) C_ A ) )
14	10, 13	sylibrd 249	. . . . 5 |- ( A e. On -> ( A ~< ( card ` B ) -> A e. ( card ` B ) ) )
15	4, 14	syl5 34	. . . 4 |- ( A e. On -> ( ( A ~< B /\ B e. dom card ) -> A e. ( card ` B ) ) )
16	15	expcomd 454	. . 3 |- ( A e. On -> ( B e. dom card -> ( A ~< B -> A e. ( card ` B ) ) ) )
17	16	imp 445	. 2 |- ( ( A e. On /\ B e. dom card ) -> ( A ~< B -> A e. ( card ` B ) ) )
18		cardsdomelir 8799	. 2 |- ( A e. ( card ` B ) -> A ~< B )
19	17, 18	impbid1 215	1 |- ( ( A e. On /\ B e. dom card ) -> ( A ~< B <-> A e. ( card ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   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-sdom 7958  df-card 8765

This theorem is referenced by:  iscard  8801  cardval2  8817  infxpenlem  8836  alephnbtwn  8894  alephnbtwn2  8895  alephord2  8899  alephsdom  8909  pwsdompw  9026  inaprc  9658