Theorem cardne 8791

Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)

Assertion

Ref	Expression
cardne	|- ( A e. ( card ` B ) -> -. A ~~ B )

Proof of Theorem cardne

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6220	. 2 |- ( A e. ( card ` B ) -> B e. dom card )
2		cardon 8770	. . . . . . . . . 10 |- ( card ` B ) e. On
3	2	oneli 5835	. . . . . . . . 9 |- ( A e. ( card ` B ) -> A e. On )
4		breq1 4656	. . . . . . . . . 10 |- ( x = A -> ( x ~~ B <-> A ~~ B ) )
5	4	onintss 5775	. . . . . . . . 9 |- ( A e. On -> ( A ~~ B -> |^| { x e. On | x ~~ B } C_ A ) )
6	3, 5	syl 17	. . . . . . . 8 |- ( A e. ( card ` B ) -> ( A ~~ B -> |^| { x e. On | x ~~ B } C_ A ) )
7	6	adantl 482	. . . . . . 7 |- ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( A ~~ B -> |^| { x e. On | x ~~ B } C_ A ) )
8		cardval3 8778	. . . . . . . . 9 |- ( B e. dom card -> ( card ` B ) = |^| { x e. On | x ~~ B } )
9	8	sseq1d 3632	. . . . . . . 8 |- ( B e. dom card -> ( ( card ` B ) C_ A <-> |^| { x e. On | x ~~ B } C_ A ) )
10	9	adantr 481	. . . . . . 7 |- ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( ( card ` B ) C_ A <-> |^| { x e. On | x ~~ B } C_ A ) )
11	7, 10	sylibrd 249	. . . . . 6 |- ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( A ~~ B -> ( card ` B ) C_ A ) )
12		ontri1 5757	. . . . . . . 8 |- ( ( ( card ` B ) e. On /\ A e. On ) -> ( ( card ` B ) C_ A <-> -. A e. ( card ` B ) ) )
13	2, 3, 12	sylancr 695	. . . . . . 7 |- ( A e. ( card ` B ) -> ( ( card ` B ) C_ A <-> -. A e. ( card ` B ) ) )
14	13	adantl 482	. . . . . 6 |- ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( ( card ` B ) C_ A <-> -. A e. ( card ` B ) ) )
15	11, 14	sylibd 229	. . . . 5 |- ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( A ~~ B -> -. A e. ( card ` B ) ) )
16	15	con2d 129	. . . 4 |- ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( A e. ( card ` B ) -> -. A ~~ B ) )
17	16	ex 450	. . 3 |- ( B e. dom card -> ( A e. ( card ` B ) -> ( A e. ( card ` B ) -> -. A ~~ B ) ) )
18	17	pm2.43d 53	. 2 |- ( B e. dom card -> ( A e. ( card ` B ) -> -. A ~~ B ) )
19	1, 18	mpcom 38	1 |- ( A e. ( card ` B ) -> -. A ~~ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   {crab 2916   C_ wss 3574   |^|cint 4475   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   ~~ cen 7952   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-fv 5896  df-en 7956  df-card 8765

This theorem is referenced by:  carden2b  8793  cardlim  8798  cardsdomelir  8799