Theorem canthp1 9476

Description: A slightly stronger form of Cantor's theorem: For 1 < n, n + 1 < 2 ^ n. Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
canthp1	|- ( 1o ~< A -> ( A +c 1o ) ~< ~P A )

Proof of Theorem canthp1

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

Step	Hyp	Ref	Expression
1		1sdom2 8159	. . . 4 |- 1o ~< 2o
2		sdomdom 7983	. . . 4 |- ( 1o ~< 2o -> 1o ~<_ 2o )
3		cdadom2 9009	. . . 4 |- ( 1o ~<_ 2o -> ( A +c 1o ) ~<_ ( A +c 2o ) )
4	1, 2, 3	mp2b 10	. . 3 |- ( A +c 1o ) ~<_ ( A +c 2o )
5		canthp1lem1 9474	. . 3 |- ( 1o ~< A -> ( A +c 2o ) ~<_ ~P A )
6		domtr 8009	. . 3 |- ( ( ( A +c 1o ) ~<_ ( A +c 2o ) /\ ( A +c 2o ) ~<_ ~P A ) -> ( A +c 1o ) ~<_ ~P A )
7	4, 5, 6	sylancr 695	. 2 |- ( 1o ~< A -> ( A +c 1o ) ~<_ ~P A )
8		fal 1490	. . 3 |- -. F.
9		ensym 8005	. . . . 5 |- ( ( A +c 1o ) ~~ ~P A -> ~P A ~~ ( A +c 1o ) )
10		bren 7964	. . . . 5 |- ( ~P A ~~ ( A +c 1o ) <-> E. f f : ~P A -1-1-onto-> ( A +c 1o ) )
11	9, 10	sylib 208	. . . 4 |- ( ( A +c 1o ) ~~ ~P A -> E. f f : ~P A -1-1-onto-> ( A +c 1o ) )
12		f1of 6137	. . . . . . . . . 10 |- ( f : ~P A -1-1-onto-> ( A +c 1o ) -> f : ~P A --> ( A +c 1o ) )
13		relsdom 7962	. . . . . . . . . . . 12 |- Rel ~<
14	13	brrelex2i 5159	. . . . . . . . . . 11 |- ( 1o ~< A -> A e. _V )
15		pwidg 4173	. . . . . . . . . . 11 |- ( A e. _V -> A e. ~P A )
16	14, 15	syl 17	. . . . . . . . . 10 |- ( 1o ~< A -> A e. ~P A )
17		ffvelrn 6357	. . . . . . . . . 10 |- ( ( f : ~P A --> ( A +c 1o ) /\ A e. ~P A ) -> ( f ` A ) e. ( A +c 1o ) )
18	12, 16, 17	syl2anr 495	. . . . . . . . 9 |- ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) -> ( f ` A ) e. ( A +c 1o ) )
19		cda1dif 8998	. . . . . . . . 9 |- ( ( f ` A ) e. ( A +c 1o ) -> ( ( A +c 1o ) \ { ( f ` A ) } ) ~~ A )
20	18, 19	syl 17	. . . . . . . 8 |- ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) -> ( ( A +c 1o ) \ { ( f ` A ) } ) ~~ A )
21		bren 7964	. . . . . . . 8 |- ( ( ( A +c 1o ) \ { ( f ` A ) } ) ~~ A <-> E. g g : ( ( A +c 1o ) \ { ( f ` A ) } ) -1-1-onto-> A )
22	20, 21	sylib 208	. . . . . . 7 |- ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) -> E. g g : ( ( A +c 1o ) \ { ( f ` A ) } ) -1-1-onto-> A )
23		simpll 790	. . . . . . . . 9 |- ( ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) /\ g : ( ( A +c 1o ) \ { ( f ` A ) } ) -1-1-onto-> A ) -> 1o ~< A )
24		simplr 792	. . . . . . . . 9 |- ( ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) /\ g : ( ( A +c 1o ) \ { ( f ` A ) } ) -1-1-onto-> A ) -> f : ~P A -1-1-onto-> ( A +c 1o ) )
25		simpr 477	. . . . . . . . 9 |- ( ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) /\ g : ( ( A +c 1o ) \ { ( f ` A ) } ) -1-1-onto-> A ) -> g : ( ( A +c 1o ) \ { ( f ` A ) } ) -1-1-onto-> A )
26		eqeq1 2626	. . . . . . . . . . . 12 |- ( w = x -> ( w = A <-> x = A ) )
27		id 22	. . . . . . . . . . . 12 |- ( w = x -> w = x )
28	26, 27	ifbieq2d 4111	. . . . . . . . . . 11 |- ( w = x -> if ( w = A , (/) , w ) = if ( x = A , (/) , x ) )
29	28	cbvmptv 4750	. . . . . . . . . 10 |- ( w e. ~P A |-> if ( w = A , (/) , w ) ) = ( x e. ~P A |-> if ( x = A , (/) , x ) )
30	29	coeq2i 5282	. . . . . . . . 9 |- ( ( g o. f ) o. ( w e. ~P A |-> if ( w = A , (/) , w ) ) ) = ( ( g o. f ) o. ( x e. ~P A |-> if ( x = A , (/) , x ) ) )
31		eqid 2622	. . . . . . . . . 10 |- { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( ( ( g o. f ) o. ( w e. ~P A |-> if ( w = A , (/) , w ) ) ) ` ( `' s " { z } ) ) = z ) ) } = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( ( ( g o. f ) o. ( w e. ~P A |-> if ( w = A , (/) , w ) ) ) ` ( `' s " { z } ) ) = z ) ) }
32	31	fpwwecbv 9466	. . . . . . . . 9 |- { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( ( ( g o. f ) o. ( w e. ~P A |-> if ( w = A , (/) , w ) ) ) ` ( `' s " { z } ) ) = z ) ) } = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( ( ( g o. f ) o. ( w e. ~P A |-> if ( w = A , (/) , w ) ) ) ` ( `' r " { y } ) ) = y ) ) }
33		eqid 2622	. . . . . . . . 9 |- U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( ( ( g o. f ) o. ( w e. ~P A |-> if ( w = A , (/) , w ) ) ) ` ( `' s " { z } ) ) = z ) ) } = U. dom { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a ( ( ( g o. f ) o. ( w e. ~P A |-> if ( w = A , (/) , w ) ) ) ` ( `' s " { z } ) ) = z ) ) }
34	23, 24, 25, 30, 32, 33	canthp1lem2 9475	. . . . . . . 8 |- -. ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) /\ g : ( ( A +c 1o ) \ { ( f ` A ) } ) -1-1-onto-> A )
35	34	pm2.21i 116	. . . . . . 7 |- ( ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) /\ g : ( ( A +c 1o ) \ { ( f ` A ) } ) -1-1-onto-> A ) -> F. )
36	22, 35	exlimddv 1863	. . . . . 6 |- ( ( 1o ~< A /\ f : ~P A -1-1-onto-> ( A +c 1o ) ) -> F. )
37	36	ex 450	. . . . 5 |- ( 1o ~< A -> ( f : ~P A -1-1-onto-> ( A +c 1o ) -> F. ) )
38	37	exlimdv 1861	. . . 4 |- ( 1o ~< A -> ( E. f f : ~P A -1-1-onto-> ( A +c 1o ) -> F. ) )
39	11, 38	syl5 34	. . 3 |- ( 1o ~< A -> ( ( A +c 1o ) ~~ ~P A -> F. ) )
40	8, 39	mtoi 190	. 2 |- ( 1o ~< A -> -. ( A +c 1o ) ~~ ~P A )
41		brsdom 7978	. 2 |- ( ( A +c 1o ) ~< ~P A <-> ( ( A +c 1o ) ~<_ ~P A /\ -. ( A +c 1o ) ~~ ~P A ) )
42	7, 40, 41	sylanbrc 698	1 |- ( 1o ~< A -> ( A +c 1o ) ~< ~P A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   F. wfal 1488   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   {copab 4712   |-> cmpt 4729   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1oc1o 7553   2oc2o 7554   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   +c ccda 8989

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  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990

This theorem is referenced by:  finngch  9477  gchcda1  9478