Theorem canthnum 9471

Description: The set of well-orderable subsets of a set A strictly dominates A. A stronger form of canth2 8113. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)

Assertion

Ref	Expression
canthnum	|- ( A e. V -> A ~< ( ~P A i^i dom card ) )

Proof of Theorem canthnum

Dummy variables f a r s x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 4850	. . . 4 |- ( A e. V -> ~P A e. _V )
2		inex1g 4801	. . . 4 |- ( ~P A e. _V -> ( ~P A i^i Fin ) e. _V )
3		infpwfidom 8851	. . . 4 |- ( ( ~P A i^i Fin ) e. _V -> A ~<_ ( ~P A i^i Fin ) )
4	1, 2, 3	3syl 18	. . 3 |- ( A e. V -> A ~<_ ( ~P A i^i Fin ) )
5		inex1g 4801	. . . . 5 |- ( ~P A e. _V -> ( ~P A i^i dom card ) e. _V )
6	1, 5	syl 17	. . . 4 |- ( A e. V -> ( ~P A i^i dom card ) e. _V )
7		finnum 8774	. . . . . 6 |- ( x e. Fin -> x e. dom card )
8	7	ssriv 3607	. . . . 5 |- Fin C_ dom card
9		sslin 3839	. . . . 5 |- ( Fin C_ dom card -> ( ~P A i^i Fin ) C_ ( ~P A i^i dom card ) )
10	8, 9	ax-mp 5	. . . 4 |- ( ~P A i^i Fin ) C_ ( ~P A i^i dom card )
11		ssdomg 8001	. . . 4 |- ( ( ~P A i^i dom card ) e. _V -> ( ( ~P A i^i Fin ) C_ ( ~P A i^i dom card ) -> ( ~P A i^i Fin ) ~<_ ( ~P A i^i dom card ) ) )
12	6, 10, 11	mpisyl 21	. . 3 |- ( A e. V -> ( ~P A i^i Fin ) ~<_ ( ~P A i^i dom card ) )
13		domtr 8009	. . 3 |- ( ( A ~<_ ( ~P A i^i Fin ) /\ ( ~P A i^i Fin ) ~<_ ( ~P A i^i dom card ) ) -> A ~<_ ( ~P A i^i dom card ) )
14	4, 12, 13	syl2anc 693	. 2 |- ( A e. V -> A ~<_ ( ~P A i^i dom card ) )
15		eqid 2622	. . . . . . 7 |- { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) }
16	15	fpwwecbv 9466	. . . . . 6 |- { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( f ` ( `' s " { z } ) ) = z ) ) }
17		eqid 2622	. . . . . 6 |- U. dom { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } = U. dom { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) }
18		eqid 2622	. . . . . 6 |- ( `' ( { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } ` U. dom { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } ) " { ( f ` U. dom { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } ) } ) = ( `' ( { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } ` U. dom { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } ) " { ( f ` U. dom { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( f ` ( `' r " { y } ) ) = y ) ) } ) } )
19	16, 17, 18	canthnumlem 9470	. . . . 5 |- ( A e. V -> -. f : ( ~P A i^i dom card ) -1-1-> A )
20		f1of1 6136	. . . . 5 |- ( f : ( ~P A i^i dom card ) -1-1-onto-> A -> f : ( ~P A i^i dom card ) -1-1-> A )
21	19, 20	nsyl 135	. . . 4 |- ( A e. V -> -. f : ( ~P A i^i dom card ) -1-1-onto-> A )
22	21	nexdv 1864	. . 3 |- ( A e. V -> -. E. f f : ( ~P A i^i dom card ) -1-1-onto-> A )
23		ensym 8005	. . . 4 |- ( A ~~ ( ~P A i^i dom card ) -> ( ~P A i^i dom card ) ~~ A )
24		bren 7964	. . . 4 |- ( ( ~P A i^i dom card ) ~~ A <-> E. f f : ( ~P A i^i dom card ) -1-1-onto-> A )
25	23, 24	sylib 208	. . 3 |- ( A ~~ ( ~P A i^i dom card ) -> E. f f : ( ~P A i^i dom card ) -1-1-onto-> A )
26	22, 25	nsyl 135	. 2 |- ( A e. V -> -. A ~~ ( ~P A i^i dom card ) )
27		brsdom 7978	. 2 |- ( A ~< ( ~P A i^i dom card ) <-> ( A ~<_ ( ~P A i^i dom card ) /\ -. A ~~ ( ~P A i^i dom card ) ) )
28	14, 26, 27	sylanbrc 698	1 |- ( A e. V -> A ~< ( ~P A i^i dom card ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   {copab 4712   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   cardccrd 8761

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-int 4476  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-om 7066  df-1st 7168  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765

This theorem is referenced by: (None)