Theorem iscard2 8802

Description: Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)

Assertion

Ref	Expression
iscard2	|- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. On ( A ~~ x -> A C_ x ) ) )

Distinct variable group:   x, A

Proof of Theorem iscard2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardon 8770	. . 3 |- ( card ` A ) e. On
2		eleq1 2689	. . 3 |- ( ( card ` A ) = A -> ( ( card ` A ) e. On <-> A e. On ) )
3	1, 2	mpbii 223	. 2 |- ( ( card ` A ) = A -> A e. On )
4		cardonle 8783	. . . . . 6 |- ( A e. On -> ( card ` A ) C_ A )
5	4	biantrurd 529	. . . . 5 |- ( A e. On -> ( A C_ ( card ` A ) <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) ) )
6		eqss 3618	. . . . 5 |- ( ( card ` A ) = A <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) )
7	5, 6	syl6rbbr 279	. . . 4 |- ( A e. On -> ( ( card ` A ) = A <-> A C_ ( card ` A ) ) )
8		oncardval 8781	. . . . 5 |- ( A e. On -> ( card ` A ) = |^| { y e. On | y ~~ A } )
9	8	sseq2d 3633	. . . 4 |- ( A e. On -> ( A C_ ( card ` A ) <-> A C_ |^| { y e. On | y ~~ A } ) )
10	7, 9	bitrd 268	. . 3 |- ( A e. On -> ( ( card ` A ) = A <-> A C_ |^| { y e. On | y ~~ A } ) )
11		ssint 4493	. . . 4 |- ( A C_ |^| { y e. On | y ~~ A } <-> A. x e. { y e. On | y ~~ A } A C_ x )
12		breq1 4656	. . . . . . . . 9 |- ( y = x -> ( y ~~ A <-> x ~~ A ) )
13	12	elrab 3363	. . . . . . . 8 |- ( x e. { y e. On | y ~~ A } <-> ( x e. On /\ x ~~ A ) )
14		ensymb 8004	. . . . . . . . 9 |- ( x ~~ A <-> A ~~ x )
15	14	anbi2i 730	. . . . . . . 8 |- ( ( x e. On /\ x ~~ A ) <-> ( x e. On /\ A ~~ x ) )
16	13, 15	bitri 264	. . . . . . 7 |- ( x e. { y e. On | y ~~ A } <-> ( x e. On /\ A ~~ x ) )
17	16	imbi1i 339	. . . . . 6 |- ( ( x e. { y e. On | y ~~ A } -> A C_ x ) <-> ( ( x e. On /\ A ~~ x ) -> A C_ x ) )
18		impexp 462	. . . . . 6 |- ( ( ( x e. On /\ A ~~ x ) -> A C_ x ) <-> ( x e. On -> ( A ~~ x -> A C_ x ) ) )
19	17, 18	bitri 264	. . . . 5 |- ( ( x e. { y e. On | y ~~ A } -> A C_ x ) <-> ( x e. On -> ( A ~~ x -> A C_ x ) ) )
20	19	ralbii2 2978	. . . 4 |- ( A. x e. { y e. On | y ~~ A } A C_ x <-> A. x e. On ( A ~~ x -> A C_ x ) )
21	11, 20	bitri 264	. . 3 |- ( A C_ |^| { y e. On | y ~~ A } <-> A. x e. On ( A ~~ x -> A C_ x ) )
22	10, 21	syl6bb 276	. 2 |- ( A e. On -> ( ( card ` A ) = A <-> A. x e. On ( A ~~ x -> A C_ x ) ) )
23	3, 22	biadan2 674	1 |- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. On ( A ~~ x -> A C_ x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   |^|cint 4475   class class class wbr 4653   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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-card 8765

This theorem is referenced by:  harcard  8804