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Theorem difsnen 7618
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
StepHypRef Expression
1 difexg 4604 . . . . . 6  |-  ( X  e.  V  ->  ( X  \  { A }
)  e.  _V )
2 enrefg 7566 . . . . . 6  |-  ( ( X  \  { A } )  e.  _V  ->  ( X  \  { A } )  ~~  ( X  \  { A }
) )
31, 2syl 16 . . . . 5  |-  ( X  e.  V  ->  ( X  \  { A }
)  ~~  ( X  \  { A } ) )
433ad2ant1 1017 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { A }
) )
5 sneq 4042 . . . . . 6  |-  ( A  =  B  ->  { A }  =  { B } )
65difeq2d 3618 . . . . 5  |-  ( A  =  B  ->  ( X  \  { A }
)  =  ( X 
\  { B }
) )
76breq2d 4468 . . . 4  |-  ( A  =  B  ->  (
( X  \  { A } )  ~~  ( X  \  { A }
)  <->  ( X  \  { A } )  ~~  ( X  \  { B } ) ) )
84, 7syl5ibcom 220 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  =  B  ->  ( X  \  { A } )  ~~  ( X  \  { B } ) ) )
98imp 429 . 2  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =  B )  ->  ( X  \  { A }
)  ~~  ( X  \  { B } ) )
10 simpl1 999 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  X  e.  V )
11 difexg 4604 . . . . . 6  |-  ( ( X  \  { A } )  e.  _V  ->  ( ( X  \  { A } )  \  { B } )  e. 
_V )
12 enrefg 7566 . . . . . 6  |-  ( ( ( X  \  { A } )  \  { B } )  e.  _V  ->  ( ( X  \  { A } )  \  { B } )  ~~  ( ( X  \  { A } )  \  { B } ) )
1310, 1, 11, 124syl 21 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  ~~  (
( X  \  { A } )  \  { B } ) )
14 dif32 3768 . . . . 5  |-  ( ( X  \  { A } )  \  { B } )  =  ( ( X  \  { B } )  \  { A } )
1513, 14syl6breq 4495 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  ~~  (
( X  \  { B } )  \  { A } ) )
16 simpl3 1001 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  e.  X )
17 simpl2 1000 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  e.  X )
18 en2sn 7614 . . . . 5  |-  ( ( B  e.  X  /\  A  e.  X )  ->  { B }  ~~  { A } )
1916, 17, 18syl2anc 661 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  { B }  ~~  { A }
)
20 incom 3687 . . . . . 6  |-  ( ( ( X  \  { A } )  \  { B } )  i^i  { B } )  =  ( { B }  i^i  ( ( X  \  { A } )  \  { B } ) )
21 disjdif 3903 . . . . . 6  |-  ( { B }  i^i  (
( X  \  { A } )  \  { B } ) )  =  (/)
2220, 21eqtri 2486 . . . . 5  |-  ( ( ( X  \  { A } )  \  { B } )  i^i  { B } )  =  (/)
2322a1i 11 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  i^i  { B } )  =  (/) )
24 incom 3687 . . . . . 6  |-  ( ( ( X  \  { B } )  \  { A } )  i^i  { A } )  =  ( { A }  i^i  ( ( X  \  { B } )  \  { A } ) )
25 disjdif 3903 . . . . . 6  |-  ( { A }  i^i  (
( X  \  { B } )  \  { A } ) )  =  (/)
2624, 25eqtri 2486 . . . . 5  |-  ( ( ( X  \  { B } )  \  { A } )  i^i  { A } )  =  (/)
2726a1i 11 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { B } )  \  { A } )  i^i  { A } )  =  (/) )
28 unen 7617 . . . 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 } ) )
2915, 19, 23, 27, 28syl22anc 1229 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  u.  { B } )  ~~  (
( ( X  \  { B } )  \  { A } )  u. 
{ A } ) )
30 simpr 461 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  =/=  B )
3130necomd 2728 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  =/=  A )
32 eldifsn 4157 . . . . 5  |-  ( B  e.  ( X  \  { A } )  <->  ( B  e.  X  /\  B  =/= 
A ) )
3316, 31, 32sylanbrc 664 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  e.  ( X  \  { A } ) )
34 difsnid 4178 . . . 4  |-  ( B  e.  ( X  \  { A } )  -> 
( ( ( X 
\  { A }
)  \  { B } )  u.  { B } )  =  ( X  \  { A } ) )
3533, 34syl 16 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  u.  { B } )  =  ( X  \  { A } ) )
36 eldifsn 4157 . . . . 5  |-  ( A  e.  ( X  \  { B } )  <->  ( A  e.  X  /\  A  =/= 
B ) )
3717, 30, 36sylanbrc 664 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  e.  ( X  \  { B } ) )
38 difsnid 4178 . . . 4  |-  ( A  e.  ( X  \  { B } )  -> 
( ( ( X 
\  { B }
)  \  { A } )  u.  { A } )  =  ( X  \  { B } ) )
3937, 38syl 16 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { B } )  \  { A } )  u.  { A } )  =  ( X  \  { B } ) )
4029, 35, 393brtr3d 4485 . 2  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( X  \  { A } ) 
~~  ( X  \  { B } ) )
419, 40pm2.61dane 2775 1  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { B }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470   (/)c0 3793   {csn 4032   class class class wbr 4456    ~~ cen 7532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-1o 7148  df-er 7329  df-en 7536
This theorem is referenced by:  domdifsn  7619  domunsncan  7636  enfixsn  7645  infdifsn  8090  cda1dif  8573
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