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Theorem difsnen 6944
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 4162 . . . . . 6  |-  ( X  e.  V  ->  ( X  \  { A }
)  e.  _V )
2 enrefg 6893 . . . . . 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 3651 . . . . . 6  |-  ( A  =  B  ->  { A }  =  { B } )
65difeq2d 3294 . . . . 5  |-  ( A  =  B  ->  ( X  \  { A }
)  =  ( X 
\  { B }
) )
76breq2d 4035 . . . 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 4162 . . . . . . 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 6893 . . . . . 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 3431 . . . . 5  |-  ( ( X  \  { A } )  \  { B } )  =  ( ( X  \  { B } )  \  { A } )
1614, 15syl6breq 4062 . . . 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 6940 . . . . 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 3361 . . . . . 6  |-  ( ( ( X  \  { A } )  \  { B } )  i^i  { B } )  =  ( { B }  i^i  ( ( X  \  { A } )  \  { B } ) )
22 disjdif 3526 . . . . . 6  |-  ( { B }  i^i  (
( X  \  { A } )  \  { B } ) )  =  (/)
2321, 22eqtri 2303 . . . . 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 3361 . . . . . 6  |-  ( ( ( X  \  { B } )  \  { A } )  i^i  { A } )  =  ( { A }  i^i  ( ( X  \  { B } )  \  { A } ) )
26 disjdif 3526 . . . . . 6  |-  ( { A }  i^i  (
( X  \  { B } )  \  { A } ) )  =  (/)
2725, 26eqtri 2303 . . . . 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 6943 . . . 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 2529 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  =/=  A )
33 eldifsn 3749 . . . . 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 3761 . . . 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 3749 . . . . 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 3761 . . . 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 4052 . 2  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( X  \  { A } ) 
~~  ( X  \  { B } ) )
429, 41pm2.61dane 2524 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   class class class wbr 4023    ~~ cen 6860
This theorem is referenced by:  domdifsn  6945  domunsncan  6962  infdifsn  7357  cda1dif  7802  enfixsn  27257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-1o 6479  df-er 6660  df-en 6864
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