Theorem canthwe 9046

Description: The set of well-orders of a set A strictly dominates A. A stronger form of canth2 7689. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)

Hypothesis

Ref	Expression
canthwe.1	|- O = { <. x , r >. | ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) }

Assertion

Ref	Expression
canthwe	|- ( A e. V -> A ~< O )

Distinct variable groups:   x, r, O   V, r, x   A, r, x

Proof of Theorem canthwe

Dummy variables u y f v w a s z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 996	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x C_ A )
2		selpw 4022	. . . . . . . 8 |- ( x e. ~P A <-> x C_ A )
3	1, 2	sylibr 212	. . . . . . 7 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x e. ~P A )
4		simp2 997	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( x X. x ) )
5		xpss12 5117	. . . . . . . . . 10 |- ( ( x C_ A /\ x C_ A ) -> ( x X. x ) C_ ( A X. A ) )
6	1, 1, 5	syl2anc 661	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> ( x X. x ) C_ ( A X. A ) )
7	4, 6	sstrd 3509	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( A X. A ) )
8		selpw 4022	. . . . . . . 8 |- ( r e. ~P ( A X. A ) <-> r C_ ( A X. A ) )
9	7, 8	sylibr 212	. . . . . . 7 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r e. ~P ( A X. A ) )
10	3, 9	jca 532	. . . . . 6 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> ( x e. ~P A /\ r e. ~P ( A X. A ) ) )
11	10	ssopab2i 4784	. . . . 5 |- { <. x , r >. | ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) } C_ { <. x , r >. | ( x e. ~P A /\ r e. ~P ( A X. A ) ) }
12		canthwe.1	. . . . 5 |- O = { <. x , r >. | ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) }
13		df-xp 5014	. . . . 5 |- ( ~P A X. ~P ( A X. A ) ) = { <. x , r >. | ( x e. ~P A /\ r e. ~P ( A X. A ) ) }
14	11, 12, 13	3sstr4i 3538	. . . 4 |- O C_ ( ~P A X. ~P ( A X. A ) )
15		pwexg 4640	. . . . 5 |- ( A e. V -> ~P A e. _V )
16		sqxpexg 6604	. . . . . 6 |- ( A e. V -> ( A X. A ) e. _V )
17		pwexg 4640	. . . . . 6 |- ( ( A X. A ) e. _V -> ~P ( A X. A ) e. _V )
18	16, 17	syl 16	. . . . 5 |- ( A e. V -> ~P ( A X. A ) e. _V )
19		xpexg 6601	. . . . 5 |- ( ( ~P A e. _V /\ ~P ( A X. A ) e. _V ) -> ( ~P A X. ~P ( A X. A ) ) e. _V )
20	15, 18, 19	syl2anc 661	. . . 4 |- ( A e. V -> ( ~P A X. ~P ( A X. A ) ) e. _V )
21		ssexg 4602	. . . 4 |- ( ( O C_ ( ~P A X. ~P ( A X. A ) ) /\ ( ~P A X. ~P ( A X. A ) ) e. _V ) -> O e. _V )
22	14, 20, 21	sylancr 663	. . 3 |- ( A e. V -> O e. _V )
23		simpr 461	. . . . . . . 8 |- ( ( A e. V /\ u e. A ) -> u e. A )
24	23	snssd 4177	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> { u } C_ A )
25		0ss 3823	. . . . . . . 8 |- (/) C_ ( { u } X. { u } )
26	25	a1i 11	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) C_ ( { u } X. { u } ) )
27		rel0 5136	. . . . . . . 8 |- Rel (/)
28		br0 4502	. . . . . . . . 9 |- -. u (/) u
29		wesn 5080	. . . . . . . . 9 |- ( Rel (/) -> ( (/) We { u } <-> -. u (/) u ) )
30	28, 29	mpbiri 233	. . . . . . . 8 |- ( Rel (/) -> (/) We { u } )
31	27, 30	mp1i 12	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) We { u } )
32		snex 4697	. . . . . . . 8 |- { u } e. _V
33		0ex 4587	. . . . . . . 8 |- (/) e. _V
34		simpl 457	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> x = { u } )
35	34	sseq1d 3526	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( x C_ A <-> { u } C_ A ) )
36		simpr 461	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> r = (/) )
37	34	sqxpeqd 5034	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> ( x X. x ) = ( { u } X. { u } ) )
38	36, 37	sseq12d 3528	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r C_ ( x X. x ) <-> (/) C_ ( { u } X. { u } ) ) )
39		weeq2 4877	. . . . . . . . . 10 |- ( x = { u } -> ( r We x <-> r We { u } ) )
40		weeq1 4876	. . . . . . . . . 10 |- ( r = (/) -> ( r We { u } <-> (/) We { u } ) )
41	39, 40	sylan9bb 699	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r We x <-> (/) We { u } ) )
42	35, 38, 41	3anbi123d 1299	. . . . . . . 8 |- ( ( x = { u } /\ r = (/) ) -> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) <-> ( { u } C_ A /\ (/) C_ ( { u } X. { u } ) /\ (/) We { u } ) ) )
43	32, 33, 42	opelopaba 4772	. . . . . . 7 |- ( <. { u } , (/) >. e. { <. x , r >. | ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) } <-> ( { u } C_ A /\ (/) C_ ( { u } X. { u } ) /\ (/) We { u } ) )
44	24, 26, 31, 43	syl3anbrc 1180	. . . . . 6 |- ( ( A e. V /\ u e. A ) -> <. { u } , (/) >. e. { <. x , r >. | ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) } )
45	44, 12	syl6eleqr 2556	. . . . 5 |- ( ( A e. V /\ u e. A ) -> <. { u } , (/) >. e. O )
46	45	ex 434	. . . 4 |- ( A e. V -> ( u e. A -> <. { u } , (/) >. e. O ) )
47		eqid 2457	. . . . . . 7 |- (/) = (/)
48		snex 4697	. . . . . . . 8 |- { v } e. _V
49	48, 33	opth2 4734	. . . . . . 7 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> ( { u } = { v } /\ (/) = (/) ) )
50	47, 49	mpbiran2 919	. . . . . 6 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> { u } = { v } )
51		vex 3112	. . . . . . 7 |- u e. _V
52		sneqbg 4202	. . . . . . 7 |- ( u e. _V -> ( { u } = { v } <-> u = v ) )
53	51, 52	ax-mp 5	. . . . . 6 |- ( { u } = { v } <-> u = v )
54	50, 53	bitri 249	. . . . 5 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> u = v )
55	54	a1ii 27	. . . 4 |- ( A e. V -> ( ( u e. A /\ v e. A ) -> ( <. { u } , (/) >. = <. { v } , (/) >. <-> u = v ) ) )
56	46, 55	dom2d 7575	. . 3 |- ( A e. V -> ( O e. _V -> A ~<_ O ) )
57	22, 56	mpd 15	. 2 |- ( A e. V -> A ~<_ O )
58		eqid 2457	. . . . . . 7 |- { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) }
59	58	fpwwe2cbv 9025	. . . . . 6 |- { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / w ]. ( w f ( r i^i ( w X. w ) ) ) = y ) ) }
60		eqid 2457	. . . . . 6 |- U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } = U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) }
61		eqid 2457	. . . . . 6 |- ( `' ( { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } ` U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } ) " { ( U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } f ( { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } ` U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } ) ) } ) = ( `' ( { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } ` U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } ) " { ( U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } f ( { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } ` U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v f ( s i^i ( v X. v ) ) ) = z ) ) } ) ) } )
62	12, 59, 60, 61	canthwelem 9045	. . . . 5 |- ( A e. V -> -. f : O -1-1-> A )
63		f1of1 5821	. . . . 5 |- ( f : O -1-1-onto-> A -> f : O -1-1-> A )
64	62, 63	nsyl 121	. . . 4 |- ( A e. V -> -. f : O -1-1-onto-> A )
65	64	nexdv 1885	. . 3 |- ( A e. V -> -. E. f f : O -1-1-onto-> A )
66		ensym 7583	. . . 4 |- ( A ~~ O -> O ~~ A )
67		bren 7544	. . . 4 |- ( O ~~ A <-> E. f f : O -1-1-onto-> A )
68	66, 67	sylib 196	. . 3 |- ( A ~~ O -> E. f f : O -1-1-onto-> A )
69	65, 68	nsyl 121	. 2 |- ( A e. V -> -. A ~~ O )
70		brsdom 7557	. 2 |- ( A ~< O <-> ( A ~<_ O /\ -. A ~~ O ) )
71	57, 69, 70	sylanbrc 664	1 |- ( A e. V -> A ~< O )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   E.wex 1613   e. wcel 1819   A.wral 2807   _Vcvv 3109   [.wsbc 3327   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   <.cop 4038   U.cuni 4251   class class class wbr 4456   {copab 4514   We wwe 4846   X. cxp 5006   `'ccnv 5007   dom cdm 5008   "cima 5011   Rel wrel 5013   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   ~~ cen 7532   ~<_ cdom 7533   ~< csdm 7534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-recs 7060  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-oi 7953

This theorem is referenced by: (None)