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Theorem cda1dif 8056
Description: Adding and subtracting one gives back the original set. Similar to pncan 9311 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
cda1dif  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { B }
)  ~~  A )

Proof of Theorem cda1dif
StepHypRef Expression
1 ovex 6106 . . . 4  |-  ( A  +c  1o )  e. 
_V
21a1i 11 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( A  +c  1o )  e. 
_V )
3 id 20 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  B  e.  ( A  +c  1o ) )
4 df1o2 6736 . . . . . . . 8  |-  1o  =  { (/) }
54xpeq1i 4898 . . . . . . 7  |-  ( 1o 
X.  { 1o }
)  =  ( {
(/) }  X.  { 1o } )
6 0ex 4339 . . . . . . . 8  |-  (/)  e.  _V
7 1on 6731 . . . . . . . . 9  |-  1o  e.  On
87elexi 2965 . . . . . . . 8  |-  1o  e.  _V
96, 8xpsn 5910 . . . . . . 7  |-  ( {
(/) }  X.  { 1o } )  =  { <.
(/) ,  1o >. }
105, 9eqtri 2456 . . . . . 6  |-  ( 1o 
X.  { 1o }
)  =  { <. (/)
,  1o >. }
11 ssun2 3511 . . . . . 6  |-  ( 1o 
X.  { 1o }
)  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
1210, 11eqsstr3i 3379 . . . . 5  |-  { <. (/)
,  1o >. }  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
13 opex 4427 . . . . . 6  |-  <. (/) ,  1o >.  e.  _V
1413snss 3926 . . . . 5  |-  ( <. (/)
,  1o >.  e.  ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  <->  { <. (/) ,  1o >. }  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
1512, 14mpbir 201 . . . 4  |-  <. (/) ,  1o >.  e.  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
16 relxp 4983 . . . . . . . 8  |-  Rel  ( _V  X.  _V )
17 cdafn 8049 . . . . . . . . . 10  |-  +c  Fn  ( _V  X.  _V )
18 fndm 5544 . . . . . . . . . 10  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
1917, 18ax-mp 8 . . . . . . . . 9  |-  dom  +c  =  ( _V  X.  _V )
2019releqi 4960 . . . . . . . 8  |-  ( Rel 
dom  +c  <->  Rel  ( _V  X.  _V ) )
2116, 20mpbir 201 . . . . . . 7  |-  Rel  dom  +c
2221ovrcl 6111 . . . . . 6  |-  ( B  e.  ( A  +c  1o )  ->  ( A  e.  _V  /\  1o  e.  _V ) )
2322simpld 446 . . . . 5  |-  ( B  e.  ( A  +c  1o )  ->  A  e. 
_V )
24 cdaval 8050 . . . . 5  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
2523, 7, 24sylancl 644 . . . 4  |-  ( B  e.  ( A  +c  1o )  ->  ( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
2615, 25syl5eleqr 2523 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
27 difsnen 7190 . . 3  |-  ( ( ( A  +c  1o )  e.  _V  /\  B  e.  ( A  +c  1o )  /\  <. (/) ,  1o >.  e.  ( A  +c  1o ) )  ->  (
( A  +c  1o )  \  { B }
)  ~~  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } ) )
282, 3, 26, 27syl3anc 1184 . 2  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { B }
)  ~~  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } ) )
2925difeq1d 3464 . . . 4  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )  \  { <. (/) ,  1o >. } ) )
30 xp01disj 6740 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
31 disj3 3672 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  i^i  ( 1o  X.  { 1o }
) )  =  (/)  <->  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) ) )
3230, 31mpbi 200 . . . . 5  |-  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) )
33 difun2 3707 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  \  ( 1o  X.  { 1o }
) )
3410difeq2i 3462 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )  \  { <. (/) ,  1o >. } )
3532, 33, 343eqtr2i 2462 . . . 4  |-  ( A  X.  { (/) } )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o 
X.  { 1o }
) )  \  { <.
(/) ,  1o >. } )
3629, 35syl6eqr 2486 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  =  ( A  X.  { (/) } ) )
37 xpsneng 7193 . . . 4  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
3823, 6, 37sylancl 644 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
3936, 38eqbrtrd 4232 . 2  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  ~~  A
)
40 entr 7159 . 2  |-  ( ( ( ( A  +c  1o )  \  { B } )  ~~  (
( A  +c  1o )  \  { <. (/) ,  1o >. } )  /\  (
( A  +c  1o )  \  { <. (/) ,  1o >. } )  ~~  A
)  ->  ( ( A  +c  1o )  \  { B } )  ~~  A )
4128, 39, 40syl2anc 643 1  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { B }
)  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814   <.cop 3817   class class class wbr 4212   Oncon0 4581    X. cxp 4876   dom cdm 4878   Rel wrel 4883    Fn wfn 5449  (class class class)co 6081   1oc1o 6717    ~~ cen 7106    +c ccda 8047
This theorem is referenced by:  canthp1  8529
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-1o 6724  df-er 6905  df-en 7110  df-cda 8048
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