MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difsnen Structured version   Unicode version

Theorem difsnen 7182
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 4343 . . . . . 6  |-  ( X  e.  V  ->  ( X  \  { A }
)  e.  _V )
2 enrefg 7131 . . . . . 6  |-  ( ( X  \  { A } )  e.  _V  ->  ( X  \  { A } )  ~~  ( X  \  { A }
) )
31, 2syl 16 . . . . 5  |-  ( X  e.  V  ->  ( X  \  { A }
)  ~~  ( X  \  { A } ) )
433ad2ant1 978 . . . 4  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { A }
) )
5 sneq 3817 . . . . . 6  |-  ( A  =  B  ->  { A }  =  { B } )
65difeq2d 3457 . . . . 5  |-  ( A  =  B  ->  ( X  \  { A }
)  =  ( X 
\  { B }
) )
76breq2d 4216 . . . 4  |-  ( A  =  B  ->  (
( X  \  { A } )  ~~  ( X  \  { A }
)  <->  ( X  \  { A } )  ~~  ( X  \  { B } ) ) )
84, 7syl5ibcom 212 . . 3  |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A  =  B  ->  ( X  \  { A } )  ~~  ( X  \  { B } ) ) )
98imp 419 . 2  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =  B )  ->  ( X  \  { A }
)  ~~  ( X  \  { B } ) )
10 simpl1 960 . . . . . . 7  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  X  e.  V )
11 difexg 4343 . . . . . . 7  |-  ( ( X  \  { A } )  e.  _V  ->  ( ( X  \  { A } )  \  { B } )  e. 
_V )
1210, 1, 113syl 19 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  e.  _V )
13 enrefg 7131 . . . . . 6  |-  ( ( ( X  \  { A } )  \  { B } )  e.  _V  ->  ( ( X  \  { A } )  \  { B } )  ~~  ( ( X  \  { A } )  \  { B } ) )
1412, 13syl 16 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  ~~  (
( X  \  { A } )  \  { B } ) )
15 dif32 3596 . . . . 5  |-  ( ( X  \  { A } )  \  { B } )  =  ( ( X  \  { B } )  \  { A } )
1614, 15syl6breq 4243 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( ( X  \  { A }
)  \  { B } )  ~~  (
( X  \  { B } )  \  { A } ) )
17 simpl3 962 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  e.  X )
18 simpl2 961 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  e.  X )
19 en2sn 7178 . . . . 5  |-  ( ( B  e.  X  /\  A  e.  X )  ->  { B }  ~~  { A } )
2017, 18, 19syl2anc 643 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  { B }  ~~  { A }
)
21 incom 3525 . . . . . 6  |-  ( ( ( X  \  { A } )  \  { B } )  i^i  { B } )  =  ( { B }  i^i  ( ( X  \  { A } )  \  { B } ) )
22 disjdif 3692 . . . . . 6  |-  ( { B }  i^i  (
( X  \  { A } )  \  { B } ) )  =  (/)
2321, 22eqtri 2455 . . . . 5  |-  ( ( ( X  \  { A } )  \  { B } )  i^i  { B } )  =  (/)
2423a1i 11 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  i^i  { B } )  =  (/) )
25 incom 3525 . . . . . 6  |-  ( ( ( X  \  { B } )  \  { A } )  i^i  { A } )  =  ( { A }  i^i  ( ( X  \  { B } )  \  { A } ) )
26 disjdif 3692 . . . . . 6  |-  ( { A }  i^i  (
( X  \  { B } )  \  { A } ) )  =  (/)
2725, 26eqtri 2455 . . . . 5  |-  ( ( ( X  \  { B } )  \  { A } )  i^i  { A } )  =  (/)
2827a1i 11 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { B } )  \  { A } )  i^i  { A } )  =  (/) )
29 unen 7181 . . . 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 1185 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  u.  { B } )  ~~  (
( ( X  \  { B } )  \  { A } )  u. 
{ A } ) )
31 simpr 448 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  =/=  B )
3231necomd 2681 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  =/=  A )
33 eldifsn 3919 . . . . 5  |-  ( B  e.  ( X  \  { A } )  <->  ( B  e.  X  /\  B  =/= 
A ) )
3417, 32, 33sylanbrc 646 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  B  e.  ( X  \  { A } ) )
35 difsnid 3936 . . . 4  |-  ( B  e.  ( X  \  { A } )  -> 
( ( ( X 
\  { A }
)  \  { B } )  u.  { B } )  =  ( X  \  { A } ) )
3634, 35syl 16 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { A } )  \  { B } )  u.  { B } )  =  ( X  \  { A } ) )
37 eldifsn 3919 . . . . 5  |-  ( A  e.  ( X  \  { B } )  <->  ( A  e.  X  /\  A  =/= 
B ) )
3818, 31, 37sylanbrc 646 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  A  e.  ( X  \  { B } ) )
39 difsnid 3936 . . . 4  |-  ( A  e.  ( X  \  { B } )  -> 
( ( ( X 
\  { B }
)  \  { A } )  u.  { A } )  =  ( X  \  { B } ) )
4038, 39syl 16 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( (
( X  \  { B } )  \  { A } )  u.  { A } )  =  ( X  \  { B } ) )
4130, 36, 403brtr3d 4233 . 2  |-  ( ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X
)  /\  A  =/=  B )  ->  ( X  \  { A } ) 
~~  ( X  \  { B } ) )
429, 41pm2.61dane 2676 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   class class class wbr 4204    ~~ cen 7098
This theorem is referenced by:  domdifsn  7183  domunsncan  7200  infdifsn  7601  cda1dif  8046  enfixsn  27189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-1o 6716  df-er 6897  df-en 7102
  Copyright terms: Public domain W3C validator