Theorem infcda1 9015

Description: An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
infcda1	|- ( om ~<_ A -> ( A +c 1o ) ~~ A )

Proof of Theorem infcda1

Step	Hyp	Ref	Expression
1		reldom 7961	. . . . . . . 8 |- Rel ~<_
2	1	brrelex2i 5159	. . . . . . 7 |- ( om ~<_ A -> A e. _V )
3		1on 7567	. . . . . . 7 |- 1o e. On
4		cdaval 8992	. . . . . . 7 |- ( ( A e. _V /\ 1o e. On ) -> ( A +c 1o ) = ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) )
5	2, 3, 4	sylancl 694	. . . . . 6 |- ( om ~<_ A -> ( A +c 1o ) = ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) )
6		df1o2 7572	. . . . . . . . 9 |- 1o = { (/) }
7	6	xpeq1i 5135	. . . . . . . 8 |- ( 1o X. { 1o } ) = ( { (/) } X. { 1o } )
8		0ex 4790	. . . . . . . . 9 |- (/) e. _V
9	3	elexi 3213	. . . . . . . . 9 |- 1o e. _V
10	8, 9	xpsn 6407	. . . . . . . 8 |- ( { (/) } X. { 1o } ) = { <. (/) , 1o >. }
11	7, 10	eqtr2i 2645	. . . . . . 7 |- { <. (/) , 1o >. } = ( 1o X. { 1o } )
12	11	a1i 11	. . . . . 6 |- ( om ~<_ A -> { <. (/) , 1o >. } = ( 1o X. { 1o } ) )
13	5, 12	difeq12d 3729	. . . . 5 |- ( om ~<_ A -> ( ( A +c 1o ) \ { <. (/) , 1o >. } ) = ( ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) \ ( 1o X. { 1o } ) ) )
14		difun2 4048	. . . . . 6 |- ( ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) \ ( 1o X. { 1o } ) ) = ( ( A X. { (/) } ) \ ( 1o X. { 1o } ) )
15		xp01disj 7576	. . . . . . 7 |- ( ( A X. { (/) } ) i^i ( 1o X. { 1o } ) ) = (/)
16		disj3 4021	. . . . . . 7 |- ( ( ( A X. { (/) } ) i^i ( 1o X. { 1o } ) ) = (/) <-> ( A X. { (/) } ) = ( ( A X. { (/) } ) \ ( 1o X. { 1o } ) ) )
17	15, 16	mpbi 220	. . . . . 6 |- ( A X. { (/) } ) = ( ( A X. { (/) } ) \ ( 1o X. { 1o } ) )
18	14, 17	eqtr4i 2647	. . . . 5 |- ( ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) \ ( 1o X. { 1o } ) ) = ( A X. { (/) } )
19	13, 18	syl6eq 2672	. . . 4 |- ( om ~<_ A -> ( ( A +c 1o ) \ { <. (/) , 1o >. } ) = ( A X. { (/) } ) )
20		cdadom3 9010	. . . . . . 7 |- ( ( A e. _V /\ 1o e. On ) -> A ~<_ ( A +c 1o ) )
21	2, 3, 20	sylancl 694	. . . . . 6 |- ( om ~<_ A -> A ~<_ ( A +c 1o ) )
22		domtr 8009	. . . . . 6 |- ( ( om ~<_ A /\ A ~<_ ( A +c 1o ) ) -> om ~<_ ( A +c 1o ) )
23	21, 22	mpdan 702	. . . . 5 |- ( om ~<_ A -> om ~<_ ( A +c 1o ) )
24		infdifsn 8554	. . . . 5 |- ( om ~<_ ( A +c 1o ) -> ( ( A +c 1o ) \ { <. (/) , 1o >. } ) ~~ ( A +c 1o ) )
25	23, 24	syl 17	. . . 4 |- ( om ~<_ A -> ( ( A +c 1o ) \ { <. (/) , 1o >. } ) ~~ ( A +c 1o ) )
26	19, 25	eqbrtrrd 4677	. . 3 |- ( om ~<_ A -> ( A X. { (/) } ) ~~ ( A +c 1o ) )
27	26	ensymd 8007	. 2 |- ( om ~<_ A -> ( A +c 1o ) ~~ ( A X. { (/) } ) )
28		xpsneng 8045	. . 3 |- ( ( A e. _V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
29	2, 8, 28	sylancl 694	. 2 |- ( om ~<_ A -> ( A X. { (/) } ) ~~ A )
30		entr 8008	. 2 |- ( ( ( A +c 1o ) ~~ ( A X. { (/) } ) /\ ( A X. { (/) } ) ~~ A ) -> ( A +c 1o ) ~~ A )
31	27, 29, 30	syl2anc 693	1 |- ( om ~<_ A -> ( A +c 1o ) ~~ A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   Oncon0 5723  (class class class)co 6650   omcom 7065   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   +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-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-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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-cda 8990

This theorem is referenced by:  pwcdaidm  9017  isfin4-3  9137  canthp1lem2  9475