Theorem canthwe 8830

Description: The set of well-orders of a set A strictly dominates A. A stronger form of canth2 7476. 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 988	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x C_ A )
2		selpw 3879	. . . . . . . 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 989	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( x X. x ) )
5		xpss12 4957	. . . . . . . . . 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 3378	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( A X. A ) )
8		selpw 3879	. . . . . . . 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 4628	. . . . 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 4858	. . . . 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 3407	. . . 4 |- O C_ ( ~P A X. ~P ( A X. A ) )
15		pwexg 4488	. . . . 5 |- ( A e. V -> ~P A e. _V )
16		xpexg 6519	. . . . . . 7 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
17	16	anidms 645	. . . . . 6 |- ( A e. V -> ( A X. A ) e. _V )
18		pwexg 4488	. . . . . 6 |- ( ( A X. A ) e. _V -> ~P ( A X. A ) e. _V )
19	17, 18	syl 16	. . . . 5 |- ( A e. V -> ~P ( A X. A ) e. _V )
20		xpexg 6519	. . . . 5 |- ( ( ~P A e. _V /\ ~P ( A X. A ) e. _V ) -> ( ~P A X. ~P ( A X. A ) ) e. _V )
21	15, 19, 20	syl2anc 661	. . . 4 |- ( A e. V -> ( ~P A X. ~P ( A X. A ) ) e. _V )
22		ssexg 4450	. . . 4 |- ( ( O C_ ( ~P A X. ~P ( A X. A ) ) /\ ( ~P A X. ~P ( A X. A ) ) e. _V ) -> O e. _V )
23	14, 21, 22	sylancr 663	. . 3 |- ( A e. V -> O e. _V )
24		simpr 461	. . . . . . . 8 |- ( ( A e. V /\ u e. A ) -> u e. A )
25	24	snssd 4030	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> { u } C_ A )
26		0ss 3678	. . . . . . . 8 |- (/) C_ ( { u } X. { u } )
27	26	a1i 11	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) C_ ( { u } X. { u } ) )
28		rel0 4976	. . . . . . . 8 |- Rel (/)
29		noel 3653	. . . . . . . . . 10 |- -. <. u , u >. e. (/)
30		df-br 4305	. . . . . . . . . 10 |- ( u (/) u <-> <. u , u >. e. (/) )
31	29, 30	mtbir 299	. . . . . . . . 9 |- -. u (/) u
32		wesn 4922	. . . . . . . . 9 |- ( Rel (/) -> ( (/) We { u } <-> -. u (/) u ) )
33	31, 32	mpbiri 233	. . . . . . . 8 |- ( Rel (/) -> (/) We { u } )
34	28, 33	mp1i 12	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) We { u } )
35		snex 4545	. . . . . . . 8 |- { u } e. _V
36		0ex 4434	. . . . . . . 8 |- (/) e. _V
37		simpl 457	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> x = { u } )
38	37	sseq1d 3395	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( x C_ A <-> { u } C_ A ) )
39		simpr 461	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> r = (/) )
40	37, 37	xpeq12d 4877	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> ( x X. x ) = ( { u } X. { u } ) )
41	39, 40	sseq12d 3397	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r C_ ( x X. x ) <-> (/) C_ ( { u } X. { u } ) ) )
42		weeq2 4721	. . . . . . . . . 10 |- ( x = { u } -> ( r We x <-> r We { u } ) )
43		weeq1 4720	. . . . . . . . . 10 |- ( r = (/) -> ( r We { u } <-> (/) We { u } ) )
44	42, 43	sylan9bb 699	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r We x <-> (/) We { u } ) )
45	38, 41, 44	3anbi123d 1289	. . . . . . . 8 |- ( ( x = { u } /\ r = (/) ) -> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) <-> ( { u } C_ A /\ (/) C_ ( { u } X. { u } ) /\ (/) We { u } ) ) )
46	35, 36, 45	opelopaba 4617	. . . . . . 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 } ) )
47	25, 27, 34, 46	syl3anbrc 1172	. . . . . 6 |- ( ( A e. V /\ u e. A ) -> <. { u } , (/) >. e. { <. x , r >. | ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) } )
48	47, 12	syl6eleqr 2534	. . . . 5 |- ( ( A e. V /\ u e. A ) -> <. { u } , (/) >. e. O )
49	48	ex 434	. . . 4 |- ( A e. V -> ( u e. A -> <. { u } , (/) >. e. O ) )
50		eqid 2443	. . . . . . 7 |- (/) = (/)
51		snex 4545	. . . . . . . 8 |- { v } e. _V
52	51, 36	opth2 4582	. . . . . . 7 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> ( { u } = { v } /\ (/) = (/) ) )
53	50, 52	mpbiran2 910	. . . . . 6 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> { u } = { v } )
54		vex 2987	. . . . . . 7 |- u e. _V
55		sneqbg 4055	. . . . . . 7 |- ( u e. _V -> ( { u } = { v } <-> u = v ) )
56	54, 55	ax-mp 5	. . . . . 6 |- ( { u } = { v } <-> u = v )
57	53, 56	bitri 249	. . . . 5 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> u = v )
58	57	a1ii 27	. . . 4 |- ( A e. V -> ( ( u e. A /\ v e. A ) -> ( <. { u } , (/) >. = <. { v } , (/) >. <-> u = v ) ) )
59	49, 58	dom2d 7362	. . 3 |- ( A e. V -> ( O e. _V -> A ~<_ O ) )
60	23, 59	mpd 15	. 2 |- ( A e. V -> A ~<_ O )
61		eqid 2443	. . . . . . 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 ) ) }
62	61	fpwwe2cbv 8809	. . . . . 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 ) ) }
63		eqid 2443	. . . . . 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 ) ) }
64		eqid 2443	. . . . . 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 ) ) } ) ) } )
65	12, 62, 63, 64	canthwelem 8829	. . . . 5 |- ( A e. V -> -. f : O -1-1-> A )
66		f1of1 5652	. . . . 5 |- ( f : O -1-1-onto-> A -> f : O -1-1-> A )
67	65, 66	nsyl 121	. . . 4 |- ( A e. V -> -. f : O -1-1-onto-> A )
68	67	nexdv 1818	. . 3 |- ( A e. V -> -. E. f f : O -1-1-onto-> A )
69		ensym 7370	. . . 4 |- ( A ~~ O -> O ~~ A )
70		bren 7331	. . . 4 |- ( O ~~ A <-> E. f f : O -1-1-onto-> A )
71	69, 70	sylib 196	. . 3 |- ( A ~~ O -> E. f f : O -1-1-onto-> A )
72	68, 71	nsyl 121	. 2 |- ( A e. V -> -. A ~~ O )
73		brsdom 7344	. 2 |- ( A ~< O <-> ( A ~<_ O /\ -. A ~~ O ) )
74	60, 72, 73	sylanbrc 664	1 |- ( A e. V -> A ~< O )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   A.wral 2727   _Vcvv 2984   [.wsbc 3198   i^i cin 3339   C_ wss 3340   (/)c0 3649   ~Pcpw 3872   {csn 3889   <.cop 3895   U.cuni 4103   class class class wbr 4304   {copab 4361   We wwe 4690   X. cxp 4850   `'ccnv 4851   dom cdm 4852   "cima 4855   Rel wrel 4857   -1-1->wf1 5427   -1-1-onto->wf1o 5429   ` cfv 5430  (class class class)co 6103   ~~ cen 7319   ~<_ cdom 7320   ~< csdm 7321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-recs 6844  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-oi 7736

This theorem is referenced by: (None)