Theorem canthwe 9094

Description: The set of well-orders of a set A strictly dominates A. A stronger form of canth2 7743. 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 1030	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x C_ A )
2		selpw 3949	. . . . . . . 8 |- ( x e. ~P A <-> x C_ A )
3	1, 2	sylibr 217	. . . . . . 7 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x e. ~P A )
4		simp2 1031	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( x X. x ) )
5		xpss12 4945	. . . . . . . . . 10 |- ( ( x C_ A /\ x C_ A ) -> ( x X. x ) C_ ( A X. A ) )
6	1, 1, 5	syl2anc 673	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> ( x X. x ) C_ ( A X. A ) )
7	4, 6	sstrd 3428	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( A X. A ) )
8		selpw 3949	. . . . . . . 8 |- ( r e. ~P ( A X. A ) <-> r C_ ( A X. A ) )
9	7, 8	sylibr 217	. . . . . . 7 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r e. ~P ( A X. A ) )
10	3, 9	jca 541	. . . . . 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 4729	. . . . 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 4845	. . . . 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 3457	. . . 4 |- O C_ ( ~P A X. ~P ( A X. A ) )
15		pwexg 4585	. . . . 5 |- ( A e. V -> ~P A e. _V )
16		sqxpexg 6615	. . . . . 6 |- ( A e. V -> ( A X. A ) e. _V )
17		pwexg 4585	. . . . . 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 6612	. . . . 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 673	. . . 4 |- ( A e. V -> ( ~P A X. ~P ( A X. A ) ) e. _V )
21		ssexg 4542	. . . 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 676	. . 3 |- ( A e. V -> O e. _V )
23		simpr 468	. . . . . . . 8 |- ( ( A e. V /\ u e. A ) -> u e. A )
24	23	snssd 4108	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> { u } C_ A )
25		0ss 3766	. . . . . . . 8 |- (/) C_ ( { u } X. { u } )
26	25	a1i 11	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) C_ ( { u } X. { u } ) )
27		rel0 4963	. . . . . . . 8 |- Rel (/)
28		br0 4442	. . . . . . . . 9 |- -. u (/) u
29		wesn 4911	. . . . . . . . 9 |- ( Rel (/) -> ( (/) We { u } <-> -. u (/) u ) )
30	28, 29	mpbiri 241	. . . . . . . 8 |- ( Rel (/) -> (/) We { u } )
31	27, 30	mp1i 13	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) We { u } )
32		snex 4641	. . . . . . . 8 |- { u } e. _V
33		0ex 4528	. . . . . . . 8 |- (/) e. _V
34		simpl 464	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> x = { u } )
35	34	sseq1d 3445	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( x C_ A <-> { u } C_ A ) )
36		simpr 468	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> r = (/) )
37	34	sqxpeqd 4865	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> ( x X. x ) = ( { u } X. { u } ) )
38	36, 37	sseq12d 3447	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r C_ ( x X. x ) <-> (/) C_ ( { u } X. { u } ) ) )
39		weeq2 4828	. . . . . . . . . 10 |- ( x = { u } -> ( r We x <-> r We { u } ) )
40		weeq1 4827	. . . . . . . . . 10 |- ( r = (/) -> ( r We { u } <-> (/) We { u } ) )
41	39, 40	sylan9bb 714	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r We x <-> (/) We { u } ) )
42	35, 38, 41	3anbi123d 1365	. . . . . . . 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 4717	. . . . . . 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 1214	. . . . . 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 2560	. . . . 5 |- ( ( A e. V /\ u e. A ) -> <. { u } , (/) >. e. O )
46	45	ex 441	. . . 4 |- ( A e. V -> ( u e. A -> <. { u } , (/) >. e. O ) )
47		eqid 2471	. . . . . . 7 |- (/) = (/)
48		snex 4641	. . . . . . . 8 |- { v } e. _V
49	48, 33	opth2 4680	. . . . . . 7 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> ( { u } = { v } /\ (/) = (/) ) )
50	47, 49	mpbiran2 933	. . . . . 6 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> { u } = { v } )
51		vex 3034	. . . . . . 7 |- u e. _V
52		sneqbg 4134	. . . . . . 7 |- ( u e. _V -> ( { u } = { v } <-> u = v ) )
53	51, 52	ax-mp 5	. . . . . 6 |- ( { u } = { v } <-> u = v )
54	50, 53	bitri 257	. . . . 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 7628	. . 3 |- ( A e. V -> ( O e. _V -> A ~<_ O ) )
57	22, 56	mpd 15	. 2 |- ( A e. V -> A ~<_ O )
58		eqid 2471	. . . . . . 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 9073	. . . . . 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 2471	. . . . . 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 2471	. . . . . 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 9093	. . . . 5 |- ( A e. V -> -. f : O -1-1-> A )
63		f1of1 5827	. . . . 5 |- ( f : O -1-1-onto-> A -> f : O -1-1-> A )
64	62, 63	nsyl 125	. . . 4 |- ( A e. V -> -. f : O -1-1-onto-> A )
65	64	nexdv 1790	. . 3 |- ( A e. V -> -. E. f f : O -1-1-onto-> A )
66		ensym 7636	. . . 4 |- ( A ~~ O -> O ~~ A )
67		bren 7596	. . . 4 |- ( O ~~ A <-> E. f f : O -1-1-onto-> A )
68	66, 67	sylib 201	. . 3 |- ( A ~~ O -> E. f f : O -1-1-onto-> A )
69	65, 68	nsyl 125	. 2 |- ( A e. V -> -. A ~~ O )
70		brsdom 7610	. 2 |- ( A ~< O <-> ( A ~<_ O /\ -. A ~~ O ) )
71	57, 69, 70	sylanbrc 677	1 |- ( A e. V -> A ~< O )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   A.wral 2756   _Vcvv 3031   [.wsbc 3255   i^i cin 3389   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395   {copab 4453   We wwe 4797   X. cxp 4837   `'ccnv 4838   dom cdm 4839   "cima 4842   Rel wrel 4844   -1-1->wf1 5586   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   ~~ cen 7584   ~<_ cdom 7585   ~< csdm 7586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-wrecs 7046  df-recs 7108  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-oi 8043

This theorem is referenced by: (None)