Theorem difsnen 8042

Description: All decrements of a set are equinumerous. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Assertion

Ref	Expression
difsnen	|- ( ( X e. V /\ A e. X /\ B e. X ) -> ( X \ { A } ) ~~ ( X \ { B } ) )

Proof of Theorem difsnen

Step	Hyp	Ref	Expression
1		difexg 4808	. . . . . 6 |- ( X e. V -> ( X \ { A } ) e. _V )
2		enrefg 7987	. . . . . 6 |- ( ( X \ { A } ) e. _V -> ( X \ { A } ) ~~ ( X \ { A } ) )
3	1, 2	syl 17	. . . . 5 |- ( X e. V -> ( X \ { A } ) ~~ ( X \ { A } ) )
4	3	3ad2ant1 1082	. . . 4 |- ( ( X e. V /\ A e. X /\ B e. X ) -> ( X \ { A } ) ~~ ( X \ { A } ) )
5		sneq 4187	. . . . . 6 |- ( A = B -> { A } = { B } )
6	5	difeq2d 3728	. . . . 5 |- ( A = B -> ( X \ { A } ) = ( X \ { B } ) )
7	6	breq2d 4665	. . . 4 |- ( A = B -> ( ( X \ { A } ) ~~ ( X \ { A } ) <-> ( X \ { A } ) ~~ ( X \ { B } ) ) )
8	4, 7	syl5ibcom 235	. . 3 |- ( ( X e. V /\ A e. X /\ B e. X ) -> ( A = B -> ( X \ { A } ) ~~ ( X \ { B } ) ) )
9	8	imp 445	. 2 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A = B ) -> ( X \ { A } ) ~~ ( X \ { B } ) )
10		simpl1 1064	. . . . . 6 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> X e. V )
11		difexg 4808	. . . . . 6 |- ( ( X \ { A } ) e. _V -> ( ( X \ { A } ) \ { B } ) e. _V )
12		enrefg 7987	. . . . . 6 |- ( ( ( X \ { A } ) \ { B } ) e. _V -> ( ( X \ { A } ) \ { B } ) ~~ ( ( X \ { A } ) \ { B } ) )
13	10, 1, 11, 12	4syl 19	. . . . 5 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> ( ( X \ { A } ) \ { B } ) ~~ ( ( X \ { A } ) \ { B } ) )
14		dif32 3891	. . . . 5 |- ( ( X \ { A } ) \ { B } ) = ( ( X \ { B } ) \ { A } )
15	13, 14	syl6breq 4694	. . . 4 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> ( ( X \ { A } ) \ { B } ) ~~ ( ( X \ { B } ) \ { A } ) )
16		simpl3 1066	. . . . 5 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> B e. X )
17		simpl2 1065	. . . . 5 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> A e. X )
18		en2sn 8037	. . . . 5 |- ( ( B e. X /\ A e. X ) -> { B } ~~ { A } )
19	16, 17, 18	syl2anc 693	. . . 4 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> { B } ~~ { A } )
20		incom 3805	. . . . . 6 |- ( ( ( X \ { A } ) \ { B } ) i^i { B } ) = ( { B } i^i ( ( X \ { A } ) \ { B } ) )
21		disjdif 4040	. . . . . 6 |- ( { B } i^i ( ( X \ { A } ) \ { B } ) ) = (/)
22	20, 21	eqtri 2644	. . . . 5 |- ( ( ( X \ { A } ) \ { B } ) i^i { B } ) = (/)
23	22	a1i 11	. . . 4 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> ( ( ( X \ { A } ) \ { B } ) i^i { B } ) = (/) )
24		incom 3805	. . . . . 6 |- ( ( ( X \ { B } ) \ { A } ) i^i { A } ) = ( { A } i^i ( ( X \ { B } ) \ { A } ) )
25		disjdif 4040	. . . . . 6 |- ( { A } i^i ( ( X \ { B } ) \ { A } ) ) = (/)
26	24, 25	eqtri 2644	. . . . 5 |- ( ( ( X \ { B } ) \ { A } ) i^i { A } ) = (/)
27	26	a1i 11	. . . 4 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> ( ( ( X \ { B } ) \ { A } ) i^i { A } ) = (/) )
28		unen 8040	. . . 4 |- ( ( ( ( ( X \ { A } ) \ { B } ) ~~ ( ( X \ { B } ) \ { A } ) /\ { B } ~~ { A } ) /\ ( ( ( ( X \ { A } ) \ { B } ) i^i { B } ) = (/) /\ ( ( ( X \ { B } ) \ { A } ) i^i { A } ) = (/) ) ) -> ( ( ( X \ { A } ) \ { B } ) u. { B } ) ~~ ( ( ( X \ { B } ) \ { A } ) u. { A } ) )
29	15, 19, 23, 27, 28	syl22anc 1327	. . 3 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> ( ( ( X \ { A } ) \ { B } ) u. { B } ) ~~ ( ( ( X \ { B } ) \ { A } ) u. { A } ) )
30		simpr 477	. . . . . 6 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> A =/= B )
31	30	necomd 2849	. . . . 5 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> B =/= A )
32		eldifsn 4317	. . . . 5 |- ( B e. ( X \ { A } ) <-> ( B e. X /\ B =/= A ) )
33	16, 31, 32	sylanbrc 698	. . . 4 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> B e. ( X \ { A } ) )
34		difsnid 4341	. . . 4 |- ( B e. ( X \ { A } ) -> ( ( ( X \ { A } ) \ { B } ) u. { B } ) = ( X \ { A } ) )
35	33, 34	syl 17	. . 3 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> ( ( ( X \ { A } ) \ { B } ) u. { B } ) = ( X \ { A } ) )
36		eldifsn 4317	. . . . 5 |- ( A e. ( X \ { B } ) <-> ( A e. X /\ A =/= B ) )
37	17, 30, 36	sylanbrc 698	. . . 4 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> A e. ( X \ { B } ) )
38		difsnid 4341	. . . 4 |- ( A e. ( X \ { B } ) -> ( ( ( X \ { B } ) \ { A } ) u. { A } ) = ( X \ { B } ) )
39	37, 38	syl 17	. . 3 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> ( ( ( X \ { B } ) \ { A } ) u. { A } ) = ( X \ { B } ) )
40	29, 35, 39	3brtr3d 4684	. 2 |- ( ( ( X e. V /\ A e. X /\ B e. X ) /\ A =/= B ) -> ( X \ { A } ) ~~ ( X \ { B } ) )
41	9, 40	pm2.61dane 2881	1 |- ( ( X e. V /\ A e. X /\ B e. X ) -> ( X \ { A } ) ~~ ( X \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   ~~ cen 7952

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-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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  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-suc 5729  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-1o 7560  df-er 7742  df-en 7956

This theorem is referenced by:  domdifsn  8043  domunsncan  8060  enfixsn  8069  infdifsn  8554  cda1dif  8998