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Theorem difsnen 6960
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 4178 . . . . . 6  |-  ( X  e.  V  ->  ( X  \  { A }
)  e.  _V )
2 enrefg 6909 . . . . . 6  |-  ( ( X  \  { A } )  e.  _V  ->  ( X  \  { A } )  ~~  ( X  \  { A }
) )
31, 2syl 15 . . . . 5  |-  ( X  e.  V  ->  ( X  \  { A }
)  ~~  ( X  \  { A } ) )
433ad2ant1 976 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { A }
) )
5 sneq 3664 . . . . . 6  |-  ( A  =  B  ->  { A }  =  { B } )
65difeq2d 3307 . . . . 5  |-  ( A  =  B  ->  ( X  \  { A }
)  =  ( X 
\  { B }
) )
76breq2d 4051 . . . 4  |-  ( A  =  B  ->  (
( X  \  { A } )  ~~  ( X  \  { A }
)  <->  ( X  \  { A } )  ~~  ( X  \  { B } ) ) )
84, 7syl5ibcom 211 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  =  B  ->  ( X  \  { A } )  ~~  ( X  \  { B } ) ) )
98imp 418 . 2  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =  B )  ->  ( X  \  { A }
)  ~~  ( X  \  { B } ) )
10 simpl1 958 . . . . . . 7  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  X  e.  V )
11 difexg 4178 . . . . . . 7  |-  ( ( X  \  { A } )  e.  _V  ->  ( ( X  \  { A } )  \  { B } )  e. 
_V )
1210, 1, 113syl 18 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  e.  _V )
13 enrefg 6909 . . . . . 6  |-  ( ( ( X  \  { A } )  \  { B } )  e.  _V  ->  ( ( X  \  { A } )  \  { B } )  ~~  ( ( X  \  { A } )  \  { B } ) )
1412, 13syl 15 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  ~~  (
( X  \  { A } )  \  { B } ) )
15 dif32 3444 . . . . 5  |-  ( ( X  \  { A } )  \  { B } )  =  ( ( X  \  { B } )  \  { A } )
1614, 15syl6breq 4078 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  ~~  (
( X  \  { B } )  \  { A } ) )
17 simpl3 960 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  e.  X )
18 simpl2 959 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  e.  X )
19 en2sn 6956 . . . . 5  |-  ( ( B  e.  X  /\  A  e.  X )  ->  { B }  ~~  { A } )
2017, 18, 19syl2anc 642 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  { B }  ~~  { A }
)
21 incom 3374 . . . . . 6  |-  ( ( ( X  \  { A } )  \  { B } )  i^i  { B } )  =  ( { B }  i^i  ( ( X  \  { A } )  \  { B } ) )
22 disjdif 3539 . . . . . 6  |-  ( { B }  i^i  (
( X  \  { A } )  \  { B } ) )  =  (/)
2321, 22eqtri 2316 . . . . 5  |-  ( ( ( X  \  { A } )  \  { B } )  i^i  { B } )  =  (/)
2423a1i 10 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  i^i  { B } )  =  (/) )
25 incom 3374 . . . . . 6  |-  ( ( ( X  \  { B } )  \  { A } )  i^i  { A } )  =  ( { A }  i^i  ( ( X  \  { B } )  \  { A } ) )
26 disjdif 3539 . . . . . 6  |-  ( { A }  i^i  (
( X  \  { B } )  \  { A } ) )  =  (/)
2725, 26eqtri 2316 . . . . 5  |-  ( ( ( X  \  { B } )  \  { A } )  i^i  { A } )  =  (/)
2827a1i 10 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { B } )  \  { A } )  i^i  { A } )  =  (/) )
29 unen 6959 . . . 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 } ) )
3016, 20, 24, 28, 29syl22anc 1183 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  u.  { B } )  ~~  (
( ( X  \  { B } )  \  { A } )  u. 
{ A } ) )
31 simpr 447 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  =/=  B )
3231necomd 2542 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  =/=  A )
33 eldifsn 3762 . . . . 5  |-  ( B  e.  ( X  \  { A } )  <->  ( B  e.  X  /\  B  =/= 
A ) )
3417, 32, 33sylanbrc 645 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  e.  ( X  \  { A } ) )
35 difsnid 3777 . . . 4  |-  ( B  e.  ( X  \  { A } )  -> 
( ( ( X 
\  { A }
)  \  { B } )  u.  { B } )  =  ( X  \  { A } ) )
3634, 35syl 15 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  u.  { B } )  =  ( X  \  { A } ) )
37 eldifsn 3762 . . . . 5  |-  ( A  e.  ( X  \  { B } )  <->  ( A  e.  X  /\  A  =/= 
B ) )
3818, 31, 37sylanbrc 645 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  e.  ( X  \  { B } ) )
39 difsnid 3777 . . . 4  |-  ( A  e.  ( X  \  { B } )  -> 
( ( ( X 
\  { B }
)  \  { A } )  u.  { A } )  =  ( X  \  { B } ) )
4038, 39syl 15 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { B } )  \  { A } )  u.  { A } )  =  ( X  \  { B } ) )
4130, 36, 403brtr3d 4068 . 2  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( X  \  { A } ) 
~~  ( X  \  { B } ) )
429, 41pm2.61dane 2537 1  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { B }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   class class class wbr 4039    ~~ cen 6876
This theorem is referenced by:  domdifsn  6961  domunsncan  6978  infdifsn  7373  cda1dif  7818  enfixsn  27360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-1o 6495  df-er 6676  df-en 6880
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