Theorem canthwe 8814

Description: The set of well-orders of a set A strictly dominates A. A stronger form of canth2 7460. 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 983	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x C_ A )
2		selpw 3864	. . . . . . . 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 984	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( x X. x ) )
5		xpss12 4941	. . . . . . . . . 10 |- ( ( x C_ A /\ x C_ A ) -> ( x X. x ) C_ ( A X. A ) )
6	1, 1, 5	syl2anc 656	. . . . . . . . 9 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> ( x X. x ) C_ ( A X. A ) )
7	4, 6	sstrd 3363	. . . . . . . 8 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> r C_ ( A X. A ) )
8		selpw 3864	. . . . . . . 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 529	. . . . . 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 4614	. . . . 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 4842	. . . . 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 3392	. . . 4 |- O C_ ( ~P A X. ~P ( A X. A ) )
15		pwexg 4473	. . . . 5 |- ( A e. V -> ~P A e. _V )
16		xpexg 6506	. . . . . . 7 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
17	16	anidms 640	. . . . . 6 |- ( A e. V -> ( A X. A ) e. _V )
18		pwexg 4473	. . . . . 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 6506	. . . . 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 656	. . . 4 |- ( A e. V -> ( ~P A X. ~P ( A X. A ) ) e. _V )
22		ssexg 4435	. . . 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 658	. . 3 |- ( A e. V -> O e. _V )
24		simpr 458	. . . . . . . 8 |- ( ( A e. V /\ u e. A ) -> u e. A )
25	24	snssd 4015	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> { u } C_ A )
26		0ss 3663	. . . . . . . 8 |- (/) C_ ( { u } X. { u } )
27	26	a1i 11	. . . . . . 7 |- ( ( A e. V /\ u e. A ) -> (/) C_ ( { u } X. { u } ) )
28		rel0 4960	. . . . . . . 8 |- Rel (/)
29		noel 3638	. . . . . . . . . 10 |- -. <. u , u >. e. (/)
30		df-br 4290	. . . . . . . . . 10 |- ( u (/) u <-> <. u , u >. e. (/) )
31	29, 30	mtbir 299	. . . . . . . . 9 |- -. u (/) u
32		wesn 4906	. . . . . . . . 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 4530	. . . . . . . 8 |- { u } e. _V
36		0ex 4419	. . . . . . . 8 |- (/) e. _V
37		simpl 454	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> x = { u } )
38	37	sseq1d 3380	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( x C_ A <-> { u } C_ A ) )
39		simpr 458	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> r = (/) )
40	37, 37	xpeq12d 4861	. . . . . . . . . 10 |- ( ( x = { u } /\ r = (/) ) -> ( x X. x ) = ( { u } X. { u } ) )
41	39, 40	sseq12d 3382	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r C_ ( x X. x ) <-> (/) C_ ( { u } X. { u } ) ) )
42		weeq2 4705	. . . . . . . . . 10 |- ( x = { u } -> ( r We x <-> r We { u } ) )
43		weeq1 4704	. . . . . . . . . 10 |- ( r = (/) -> ( r We { u } <-> (/) We { u } ) )
44	42, 43	sylan9bb 694	. . . . . . . . 9 |- ( ( x = { u } /\ r = (/) ) -> ( r We x <-> (/) We { u } ) )
45	38, 41, 44	3anbi123d 1284	. . . . . . . 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 4603	. . . . . . 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 1167	. . . . . 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 2532	. . . . 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 2441	. . . . . . 7 |- (/) = (/)
51		snex 4530	. . . . . . . 8 |- { v } e. _V
52	51, 36	opth2 4567	. . . . . . 7 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> ( { u } = { v } /\ (/) = (/) ) )
53	50, 52	mpbiran2 905	. . . . . 6 |- ( <. { u } , (/) >. = <. { v } , (/) >. <-> { u } = { v } )
54		vex 2973	. . . . . . 7 |- u e. _V
55		sneqbg 4040	. . . . . . 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 7346	. . 3 |- ( A e. V -> ( O e. _V -> A ~<_ O ) )
60	23, 59	mpd 15	. 2 |- ( A e. V -> A ~<_ O )
61		eqid 2441	. . . . . . 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 8793	. . . . . 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 2441	. . . . . 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 2441	. . . . . 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 8813	. . . . 5 |- ( A e. V -> -. f : O -1-1-> A )
66		f1of1 5637	. . . . 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 1939	. . 3 |- ( A e. V -> -. E. f f : O -1-1-onto-> A )
69		ensym 7354	. . . 4 |- ( A ~~ O -> O ~~ A )
70		bren 7315	. . . 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 7328	. 2 |- ( A ~< O <-> ( A ~<_ O /\ -. A ~~ O ) )
74	60, 72, 73	sylanbrc 659	1 |- ( A e. V -> A ~< O )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   = wceq 1364   E.wex 1591   e. wcel 1761   A.wral 2713   _Vcvv 2970   [.wsbc 3183   i^i cin 3324   C_ wss 3325   (/)c0 3634   ~Pcpw 3857   {csn 3874   <.cop 3880   U.cuni 4088   class class class wbr 4289   {copab 4346   We wwe 4674   X. cxp 4834   `'ccnv 4835   dom cdm 4836   "cima 4839   Rel wrel 4841   -1-1->wf1 5412   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   ~~ cen 7303   ~<_ cdom 7304   ~< csdm 7305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-recs 6828  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-oi 7720

This theorem is referenced by: (None)