Theorem tskwe 8776

Description: A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
tskwe	|- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> A e. dom card )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem tskwe

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 4850	. . . 4 |- ( A e. V -> ~P A e. _V )
2		rabexg 4812	. . . 4 |- ( ~P A e. _V -> { x e. ~P A | x ~< A } e. _V )
3		incom 3805	. . . . 5 |- ( { x e. ~P A | x ~< A } i^i On ) = ( On i^i { x e. ~P A | x ~< A } )
4		inex1g 4801	. . . . 5 |- ( { x e. ~P A | x ~< A } e. _V -> ( { x e. ~P A | x ~< A } i^i On ) e. _V )
5	3, 4	syl5eqelr 2706	. . . 4 |- ( { x e. ~P A | x ~< A } e. _V -> ( On i^i { x e. ~P A | x ~< A } ) e. _V )
6		inss1 3833	. . . . . . . . . . 11 |- ( On i^i { x e. ~P A | x ~< A } ) C_ On
7	6	sseli 3599	. . . . . . . . . 10 |- ( z e. ( On i^i { x e. ~P A | x ~< A } ) -> z e. On )
8		onelon 5748	. . . . . . . . . . 11 |- ( ( z e. On /\ y e. z ) -> y e. On )
9	8	ancoms 469	. . . . . . . . . 10 |- ( ( y e. z /\ z e. On ) -> y e. On )
10	7, 9	sylan2 491	. . . . . . . . 9 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y e. On )
11		onelss 5766	. . . . . . . . . . . . . 14 |- ( z e. On -> ( y e. z -> y C_ z ) )
12	11	impcom 446	. . . . . . . . . . . . 13 |- ( ( y e. z /\ z e. On ) -> y C_ z )
13	7, 12	sylan2 491	. . . . . . . . . . . 12 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y C_ z )
14		inss2 3834	. . . . . . . . . . . . . . . . 17 |- ( On i^i { x e. ~P A | x ~< A } ) C_ { x e. ~P A | x ~< A }
15	14	sseli 3599	. . . . . . . . . . . . . . . 16 |- ( z e. ( On i^i { x e. ~P A | x ~< A } ) -> z e. { x e. ~P A | x ~< A } )
16		breq1 4656	. . . . . . . . . . . . . . . . 17 |- ( x = z -> ( x ~< A <-> z ~< A ) )
17	16	elrab 3363	. . . . . . . . . . . . . . . 16 |- ( z e. { x e. ~P A | x ~< A } <-> ( z e. ~P A /\ z ~< A ) )
18	15, 17	sylib 208	. . . . . . . . . . . . . . 15 |- ( z e. ( On i^i { x e. ~P A | x ~< A } ) -> ( z e. ~P A /\ z ~< A ) )
19	18	simpld 475	. . . . . . . . . . . . . 14 |- ( z e. ( On i^i { x e. ~P A | x ~< A } ) -> z e. ~P A )
20	19	elpwid 4170	. . . . . . . . . . . . 13 |- ( z e. ( On i^i { x e. ~P A | x ~< A } ) -> z C_ A )
21	20	adantl 482	. . . . . . . . . . . 12 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> z C_ A )
22	13, 21	sstrd 3613	. . . . . . . . . . 11 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y C_ A )
23		selpw 4165	. . . . . . . . . . 11 |- ( y e. ~P A <-> y C_ A )
24	22, 23	sylibr 224	. . . . . . . . . 10 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y e. ~P A )
25		vex 3203	. . . . . . . . . . . 12 |- z e. _V
26		ssdomg 8001	. . . . . . . . . . . 12 |- ( z e. _V -> ( y C_ z -> y ~<_ z ) )
27	25, 13, 26	mpsyl 68	. . . . . . . . . . 11 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y ~<_ z )
28	18	simprd 479	. . . . . . . . . . . 12 |- ( z e. ( On i^i { x e. ~P A | x ~< A } ) -> z ~< A )
29	28	adantl 482	. . . . . . . . . . 11 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> z ~< A )
30		domsdomtr 8095	. . . . . . . . . . 11 |- ( ( y ~<_ z /\ z ~< A ) -> y ~< A )
31	27, 29, 30	syl2anc 693	. . . . . . . . . 10 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y ~< A )
32		breq1 4656	. . . . . . . . . . 11 |- ( x = y -> ( x ~< A <-> y ~< A ) )
33	32	elrab 3363	. . . . . . . . . 10 |- ( y e. { x e. ~P A | x ~< A } <-> ( y e. ~P A /\ y ~< A ) )
34	24, 31, 33	sylanbrc 698	. . . . . . . . 9 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y e. { x e. ~P A | x ~< A } )
35	10, 34	elind 3798	. . . . . . . 8 |- ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y e. ( On i^i { x e. ~P A | x ~< A } ) )
36	35	gen2 1723	. . . . . . 7 |- A. y A. z ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y e. ( On i^i { x e. ~P A | x ~< A } ) )
37		dftr2 4754	. . . . . . 7 |- ( Tr ( On i^i { x e. ~P A | x ~< A } ) <-> A. y A. z ( ( y e. z /\ z e. ( On i^i { x e. ~P A | x ~< A } ) ) -> y e. ( On i^i { x e. ~P A | x ~< A } ) ) )
38	36, 37	mpbir 221	. . . . . 6 |- Tr ( On i^i { x e. ~P A | x ~< A } )
39		ordon 6982	. . . . . 6 |- Ord On
40		trssord 5740	. . . . . 6 |- ( ( Tr ( On i^i { x e. ~P A | x ~< A } ) /\ ( On i^i { x e. ~P A | x ~< A } ) C_ On /\ Ord On ) -> Ord ( On i^i { x e. ~P A | x ~< A } ) )
41	38, 6, 39, 40	mp3an 1424	. . . . 5 |- Ord ( On i^i { x e. ~P A | x ~< A } )
42		elong 5731	. . . . 5 |- ( ( On i^i { x e. ~P A | x ~< A } ) e. _V -> ( ( On i^i { x e. ~P A | x ~< A } ) e. On <-> Ord ( On i^i { x e. ~P A | x ~< A } ) ) )
43	41, 42	mpbiri 248	. . . 4 |- ( ( On i^i { x e. ~P A | x ~< A } ) e. _V -> ( On i^i { x e. ~P A | x ~< A } ) e. On )
44	1, 2, 5, 43	4syl 19	. . 3 |- ( A e. V -> ( On i^i { x e. ~P A | x ~< A } ) e. On )
45	44	adantr 481	. 2 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> ( On i^i { x e. ~P A | x ~< A } ) e. On )
46		simpr 477	. . . . 5 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> { x e. ~P A | x ~< A } C_ A )
47	14, 46	syl5ss 3614	. . . 4 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> ( On i^i { x e. ~P A | x ~< A } ) C_ A )
48		ssdomg 8001	. . . . 5 |- ( A e. V -> ( ( On i^i { x e. ~P A | x ~< A } ) C_ A -> ( On i^i { x e. ~P A | x ~< A } ) ~<_ A ) )
49	48	adantr 481	. . . 4 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> ( ( On i^i { x e. ~P A | x ~< A } ) C_ A -> ( On i^i { x e. ~P A | x ~< A } ) ~<_ A ) )
50	47, 49	mpd 15	. . 3 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> ( On i^i { x e. ~P A | x ~< A } ) ~<_ A )
51		ordirr 5741	. . . . 5 |- ( Ord ( On i^i { x e. ~P A | x ~< A } ) -> -. ( On i^i { x e. ~P A | x ~< A } ) e. ( On i^i { x e. ~P A | x ~< A } ) )
52	41, 51	mp1i 13	. . . 4 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> -. ( On i^i { x e. ~P A | x ~< A } ) e. ( On i^i { x e. ~P A | x ~< A } ) )
53	44	3ad2ant1 1082	. . . . . 6 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A /\ ( On i^i { x e. ~P A | x ~< A } ) ~< A ) -> ( On i^i { x e. ~P A | x ~< A } ) e. On )
54		elpw2g 4827	. . . . . . . . . 10 |- ( A e. V -> ( ( On i^i { x e. ~P A | x ~< A } ) e. ~P A <-> ( On i^i { x e. ~P A | x ~< A } ) C_ A ) )
55	54	adantr 481	. . . . . . . . 9 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> ( ( On i^i { x e. ~P A | x ~< A } ) e. ~P A <-> ( On i^i { x e. ~P A | x ~< A } ) C_ A ) )
56	47, 55	mpbird 247	. . . . . . . 8 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> ( On i^i { x e. ~P A | x ~< A } ) e. ~P A )
57	56	3adant3 1081	. . . . . . 7 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A /\ ( On i^i { x e. ~P A | x ~< A } ) ~< A ) -> ( On i^i { x e. ~P A | x ~< A } ) e. ~P A )
58		simp3 1063	. . . . . . 7 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A /\ ( On i^i { x e. ~P A | x ~< A } ) ~< A ) -> ( On i^i { x e. ~P A | x ~< A } ) ~< A )
59		nfcv 2764	. . . . . . . . 9 |- F/_ x On
60		nfrab1 3122	. . . . . . . . 9 |- F/_ x { x e. ~P A | x ~< A }
61	59, 60	nfin 3820	. . . . . . . 8 |- F/_ x ( On i^i { x e. ~P A | x ~< A } )
62		nfcv 2764	. . . . . . . 8 |- F/_ x ~P A
63		nfcv 2764	. . . . . . . . 9 |- F/_ x ~<
64		nfcv 2764	. . . . . . . . 9 |- F/_ x A
65	61, 63, 64	nfbr 4699	. . . . . . . 8 |- F/ x ( On i^i { x e. ~P A | x ~< A } ) ~< A
66		breq1 4656	. . . . . . . 8 |- ( x = ( On i^i { x e. ~P A | x ~< A } ) -> ( x ~< A <-> ( On i^i { x e. ~P A | x ~< A } ) ~< A ) )
67	61, 62, 65, 66	elrabf 3360	. . . . . . 7 |- ( ( On i^i { x e. ~P A | x ~< A } ) e. { x e. ~P A | x ~< A } <-> ( ( On i^i { x e. ~P A | x ~< A } ) e. ~P A /\ ( On i^i { x e. ~P A | x ~< A } ) ~< A ) )
68	57, 58, 67	sylanbrc 698	. . . . . 6 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A /\ ( On i^i { x e. ~P A | x ~< A } ) ~< A ) -> ( On i^i { x e. ~P A | x ~< A } ) e. { x e. ~P A | x ~< A } )
69	53, 68	elind 3798	. . . . 5 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A /\ ( On i^i { x e. ~P A | x ~< A } ) ~< A ) -> ( On i^i { x e. ~P A | x ~< A } ) e. ( On i^i { x e. ~P A | x ~< A } ) )
70	69	3expia 1267	. . . 4 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> ( ( On i^i { x e. ~P A | x ~< A } ) ~< A -> ( On i^i { x e. ~P A | x ~< A } ) e. ( On i^i { x e. ~P A | x ~< A } ) ) )
71	52, 70	mtod 189	. . 3 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> -. ( On i^i { x e. ~P A | x ~< A } ) ~< A )
72		bren2 7986	. . 3 |- ( ( On i^i { x e. ~P A | x ~< A } ) ~~ A <-> ( ( On i^i { x e. ~P A | x ~< A } ) ~<_ A /\ -. ( On i^i { x e. ~P A | x ~< A } ) ~< A ) )
73	50, 71, 72	sylanbrc 698	. 2 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> ( On i^i { x e. ~P A | x ~< A } ) ~~ A )
74		isnumi 8772	. 2 |- ( ( ( On i^i { x e. ~P A | x ~< A } ) e. On /\ ( On i^i { x e. ~P A | x ~< A } ) ~~ A ) -> A e. dom card )
75	45, 73, 74	syl2anc 693	1 |- ( ( A e. V /\ { x e. ~P A | x ~< A } C_ A ) -> A e. dom card )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   Tr wtr 4752   dom cdm 5114   Ord word 5722   Oncon0 5723   ~~ 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-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765

This theorem is referenced by:  tskwe2  9595  grothac  9652