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Theorem difsnen 7391
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 4438 . . . . . 6  |-  ( X  e.  V  ->  ( X  \  { A }
)  e.  _V )
2 enrefg 7339 . . . . . 6  |-  ( ( X  \  { A } )  e.  _V  ->  ( X  \  { A } )  ~~  ( X  \  { A }
) )
31, 2syl 16 . . . . 5  |-  ( X  e.  V  ->  ( X  \  { A }
)  ~~  ( X  \  { A } ) )
433ad2ant1 1009 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { A }
) )
5 sneq 3885 . . . . . 6  |-  ( A  =  B  ->  { A }  =  { B } )
65difeq2d 3472 . . . . 5  |-  ( A  =  B  ->  ( X  \  { A }
)  =  ( X 
\  { B }
) )
76breq2d 4302 . . . 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 991 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  X  e.  V )
11 difexg 4438 . . . . . 6  |-  ( ( X  \  { A } )  e.  _V  ->  ( ( X  \  { A } )  \  { B } )  e. 
_V )
12 enrefg 7339 . . . . . 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 3611 . . . . 5  |-  ( ( X  \  { A } )  \  { B } )  =  ( ( X  \  { B } )  \  { A } )
1513, 14syl6breq 4329 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  ~~  (
( X  \  { B } )  \  { A } ) )
16 simpl3 993 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  e.  X )
17 simpl2 992 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  e.  X )
18 en2sn 7387 . . . . 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 3541 . . . . . 6  |-  ( ( ( X  \  { A } )  \  { B } )  i^i  { B } )  =  ( { B }  i^i  ( ( X  \  { A } )  \  { B } ) )
21 disjdif 3749 . . . . . 6  |-  ( { B }  i^i  (
( X  \  { A } )  \  { B } ) )  =  (/)
2220, 21eqtri 2461 . . . . 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 3541 . . . . . 6  |-  ( ( ( X  \  { B } )  \  { A } )  i^i  { A } )  =  ( { A }  i^i  ( ( X  \  { B } )  \  { A } ) )
25 disjdif 3749 . . . . . 6  |-  ( { A }  i^i  (
( X  \  { B } )  \  { A } ) )  =  (/)
2624, 25eqtri 2461 . . . . 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 7390 . . . 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 1219 . . 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 2693 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  =/=  A )
32 eldifsn 3998 . . . . 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 4017 . . . 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 3998 . . . . 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 4017 . . . 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 4319 . 2  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( X  \  { A } ) 
~~  ( X  \  { B } ) )
419, 40pm2.61dane 2687 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 965    = wceq 1369    e. wcel 1756    =/= wne 2604   _Vcvv 2970    \ cdif 3323    u. cun 3324    i^i cin 3325   (/)c0 3635   {csn 3875   class class class wbr 4290    ~~ cen 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-1o 6918  df-er 7099  df-en 7309
This theorem is referenced by:  domdifsn  7392  domunsncan  7409  enfixsn  7418  infdifsn  7860  cda1dif  8343
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