Theorem canthwe 9473

Description: The set of well-orders of a set A strictly dominates A. A stronger form of canth2 8113. 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 1061	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x C_ A )
2		selpw 4165	. . . . . . . 8 |- ( x e. ~P A <-> x C_ A )
3	1, 2	sylibr 224	. . . . . . 7 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x e. ~P A )
4		simp2 1062	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( x X. x ) )
5		xpss12 5225	. . . . . . . . . 10 |- ( ( x C_ A /\ x C_ A ) -> ( x X. x ) C_ ( A X. A ) )
6	1, 1, 5	syl2anc 693	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> ( x X. x ) C_ ( A X. A ) )
7	4, 6	sstrd 3613	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( A X. A ) )
8		selpw 4165	. . . . . . . 8 |- ( r e. ~P ( A X. A ) <-> r C_ ( A X. A ) )
9	7, 8	sylibr 224	. . . . . . 7 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r e. ~P ( A X. A ) )
10	3, 9	jca 554	. . . . . 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 5003	. . . . 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 5120	. . . . 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 3644	. . . 4 |- O C_ ( ~P A X. ~P ( A X. A ) )
15		pwexg 4850	. . . . 5 |- ( A e. V -> ~P A e. _V )
16		sqxpexg 6963	. . . . . 6 |- ( A e. V -> ( A X. A ) e. _V )
17		pwexg 4850	. . . . . 6 |- ( ( A X. A ) e. _V -> ~P ( A X. A ) e. _V )
18	16, 17	syl 17	. . . . 5 |- ( A e. V -> ~P ( A X. A ) e. _V )
19		xpexg 6960	. . . . 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 693	. . . 4 |- ( A e. V -> ( ~P A X. ~P ( A X. A ) ) e. _V )
21		ssexg 4804	. . . 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 695	. . 3 |- ( A e. V -> O e. _V )
23		simpr 477	. . . . . . . 8 |- ( ( A e. V /\ u e. A ) -> u e. A )
24	23	snssd 4340	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> { u } C_ A )
25		0ss 3972	. . . . . . . 8 |- (/) C_ ( { u } X. { u } )
26	25	a1i 11	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) C_ ( { u } X. { u } ) )
27		rel0 5243	. . . . . . . 8 |- Rel (/)
28		br0 4701	. . . . . . . . 9 |- -. u (/) u
29		wesn 5190	. . . . . . . . 9 |- ( Rel (/) -> ( (/) We { u } <-> -. u (/) u ) )
30	28, 29	mpbiri 248	. . . . . . . 8 |- ( Rel (/) -> (/) We { u } )
31	27, 30	mp1i 13	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) We { u } )
32		snex 4908	. . . . . . . 8 |- { u } e. _V
33		0ex 4790	. . . . . . . 8 |- (/) e. _V
34		simpl 473	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> x = { u } )
35	34	sseq1d 3632	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( x C_ A <-> { u } C_ A ) )
36		simpr 477	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> r = (/) )
37	34	sqxpeqd 5141	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> ( x X. x ) = ( { u } X. { u } ) )
38	36, 37	sseq12d 3634	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r C_ ( x X. x ) <-> (/) C_ ( { u } X. { u } ) ) )
39		weeq2 5103	. . . . . . . . . 10 |- ( x = { u } -> ( r We x <-> r We { u } ) )
40		weeq1 5102	. . . . . . . . . 10 |- ( r = (/) -> ( r We { u } <-> (/) We { u } ) )
41	39, 40	sylan9bb 736	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r We x <-> (/) We { u } ) )
42	35, 38, 41	3anbi123d 1399	. . . . . . . 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 4991	. . . . . . 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 1246	. . . . . 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 2712	. . . . 5 |- ( ( A e. V /\ u e. A ) -> <. { u } , (/) >. e. O )
46	45	ex 450	. . . 4 |- ( A e. V -> ( u e. A -> <. { u } , (/) >. e. O ) )
47		eqid 2622	. . . . . . 7 |- (/) = (/)
48		snex 4908	. . . . . . . 8 |- { v } e. _V
49	48, 33	opth2 4949	. . . . . . 7 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> ( { u } = { v } /\ (/) = (/) ) )
50	47, 49	mpbiran2 954	. . . . . 6 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> { u } = { v } )
51		vex 3203	. . . . . . 7 |- u e. _V
52		sneqbg 4374	. . . . . . 7 |- ( u e. _V -> ( { u } = { v } <-> u = v ) )
53	51, 52	ax-mp 5	. . . . . 6 |- ( { u } = { v } <-> u = v )
54	50, 53	bitri 264	. . . . 5 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> u = v )
55	54	2a1i 12	. . . 4 |- ( A e. V -> ( ( u e. A /\ v e. A ) -> ( <. { u } , (/) >. = <. { v } , (/) >. <-> u = v ) ) )
56	46, 55	dom2d 7996	. . 3 |- ( A e. V -> ( O e. _V -> A ~<_ O ) )
57	22, 56	mpd 15	. 2 |- ( A e. V -> A ~<_ O )
58		eqid 2622	. . . . . . 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 9452	. . . . . 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 2622	. . . . . 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 2622	. . . . . 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 9472	. . . . 5 |- ( A e. V -> -. f : O -1-1-> A )
63		f1of1 6136	. . . . 5 |- ( f : O -1-1-onto-> A -> f : O -1-1-> A )
64	62, 63	nsyl 135	. . . 4 |- ( A e. V -> -. f : O -1-1-onto-> A )
65	64	nexdv 1864	. . 3 |- ( A e. V -> -. E. f f : O -1-1-onto-> A )
66		ensym 8005	. . . 4 |- ( A ~~ O -> O ~~ A )
67		bren 7964	. . . 4 |- ( O ~~ A <-> E. f f : O -1-1-onto-> A )
68	66, 67	sylib 208	. . 3 |- ( A ~~ O -> E. f f : O -1-1-onto-> A )
69	65, 68	nsyl 135	. 2 |- ( A e. V -> -. A ~~ O )
70		brsdom 7978	. 2 |- ( A ~< O <-> ( A ~<_ O /\ -. A ~~ O ) )
71	57, 69, 70	sylanbrc 698	1 |- ( A e. V -> A ~< O )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   {copab 4712   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Rel wrel 5119   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954

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-rep 4771  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-se 5074  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415

This theorem is referenced by: (None)