Theorem cda1dif 8998

Description: Adding and subtracting one gives back the original set. Similar to pncan 10287 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
cda1dif	|- ( B e. ( A +c 1o ) -> ( ( A +c 1o ) \ { B } ) ~~ A )

Proof of Theorem cda1dif

Step	Hyp	Ref	Expression
1		ovexd 6680	. . 3 |- ( B e. ( A +c 1o ) -> ( A +c 1o ) e. _V )
2		id 22	. . 3 |- ( B e. ( A +c 1o ) -> B e. ( A +c 1o ) )
3		df1o2 7572	. . . . . . . 8 |- 1o = { (/) }
4	3	xpeq1i 5135	. . . . . . 7 |- ( 1o X. { 1o } ) = ( { (/) } X. { 1o } )
5		0ex 4790	. . . . . . . 8 |- (/) e. _V
6		1on 7567	. . . . . . . . 9 |- 1o e. On
7	6	elexi 3213	. . . . . . . 8 |- 1o e. _V
8	5, 7	xpsn 6407	. . . . . . 7 |- ( { (/) } X. { 1o } ) = { <. (/) , 1o >. }
9	4, 8	eqtri 2644	. . . . . 6 |- ( 1o X. { 1o } ) = { <. (/) , 1o >. }
10		ssun2 3777	. . . . . 6 |- ( 1o X. { 1o } ) C_ ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) )
11	9, 10	eqsstr3i 3636	. . . . 5 |- { <. (/) , 1o >. } C_ ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) )
12		opex 4932	. . . . . 6 |- <. (/) , 1o >. e. _V
13	12	snss 4316	. . . . 5 |- ( <. (/) , 1o >. e. ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) <-> { <. (/) , 1o >. } C_ ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) )
14	11, 13	mpbir 221	. . . 4 |- <. (/) , 1o >. e. ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) )
15		relxp 5227	. . . . . . . 8 |- Rel ( _V X. _V )
16		cdafn 8991	. . . . . . . . . 10 |- +c Fn ( _V X. _V )
17		fndm 5990	. . . . . . . . . 10 |- ( +c Fn ( _V X. _V ) -> dom +c = ( _V X. _V ) )
18	16, 17	ax-mp 5	. . . . . . . . 9 |- dom +c = ( _V X. _V )
19	18	releqi 5202	. . . . . . . 8 |- ( Rel dom +c <-> Rel ( _V X. _V ) )
20	15, 19	mpbir 221	. . . . . . 7 |- Rel dom +c
21	20	ovrcl 6686	. . . . . 6 |- ( B e. ( A +c 1o ) -> ( A e. _V /\ 1o e. _V ) )
22	21	simpld 475	. . . . 5 |- ( B e. ( A +c 1o ) -> A e. _V )
23		cdaval 8992	. . . . 5 |- ( ( A e. _V /\ 1o e. On ) -> ( A +c 1o ) = ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) )
24	22, 6, 23	sylancl 694	. . . 4 |- ( B e. ( A +c 1o ) -> ( A +c 1o ) = ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) )
25	14, 24	syl5eleqr 2708	. . 3 |- ( B e. ( A +c 1o ) -> <. (/) , 1o >. e. ( A +c 1o ) )
26		difsnen 8042	. . 3 |- ( ( ( A +c 1o ) e. _V /\ B e. ( A +c 1o ) /\ <. (/) , 1o >. e. ( A +c 1o ) ) -> ( ( A +c 1o ) \ { B } ) ~~ ( ( A +c 1o ) \ { <. (/) , 1o >. } ) )
27	1, 2, 25, 26	syl3anc 1326	. 2 |- ( B e. ( A +c 1o ) -> ( ( A +c 1o ) \ { B } ) ~~ ( ( A +c 1o ) \ { <. (/) , 1o >. } ) )
28	24	difeq1d 3727	. . . 4 |- ( B e. ( A +c 1o ) -> ( ( A +c 1o ) \ { <. (/) , 1o >. } ) = ( ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) \ { <. (/) , 1o >. } ) )
29		xp01disj 7576	. . . . . 6 |- ( ( A X. { (/) } ) i^i ( 1o X. { 1o } ) ) = (/)
30		disj3 4021	. . . . . 6 |- ( ( ( A X. { (/) } ) i^i ( 1o X. { 1o } ) ) = (/) <-> ( A X. { (/) } ) = ( ( A X. { (/) } ) \ ( 1o X. { 1o } ) ) )
31	29, 30	mpbi 220	. . . . 5 |- ( A X. { (/) } ) = ( ( A X. { (/) } ) \ ( 1o X. { 1o } ) )
32		difun2 4048	. . . . 5 |- ( ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) \ ( 1o X. { 1o } ) ) = ( ( A X. { (/) } ) \ ( 1o X. { 1o } ) )
33	9	difeq2i 3725	. . . . 5 |- ( ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) \ ( 1o X. { 1o } ) ) = ( ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) \ { <. (/) , 1o >. } )
34	31, 32, 33	3eqtr2i 2650	. . . 4 |- ( A X. { (/) } ) = ( ( ( A X. { (/) } ) u. ( 1o X. { 1o } ) ) \ { <. (/) , 1o >. } )
35	28, 34	syl6eqr 2674	. . 3 |- ( B e. ( A +c 1o ) -> ( ( A +c 1o ) \ { <. (/) , 1o >. } ) = ( A X. { (/) } ) )
36		xpsneng 8045	. . . 4 |- ( ( A e. _V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
37	22, 5, 36	sylancl 694	. . 3 |- ( B e. ( A +c 1o ) -> ( A X. { (/) } ) ~~ A )
38	35, 37	eqbrtrd 4675	. 2 |- ( B e. ( A +c 1o ) -> ( ( A +c 1o ) \ { <. (/) , 1o >. } ) ~~ A )
39		entr 8008	. 2 |- ( ( ( ( A +c 1o ) \ { B } ) ~~ ( ( A +c 1o ) \ { <. (/) , 1o >. } ) /\ ( ( A +c 1o ) \ { <. (/) , 1o >. } ) ~~ A ) -> ( ( A +c 1o ) \ { B } ) ~~ A )
40	27, 38, 39	syl2anc 693	1 |- ( B e. ( A +c 1o ) -> ( ( A +c 1o ) \ { B } ) ~~ A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Rel wrel 5119   Oncon0 5723   Fn wfn 5883  (class class class)co 6650   1oc1o 7553   ~~ cen 7952   +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-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-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-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-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-cda 8990

This theorem is referenced by:  canthp1  9476