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Theorem cda1dif 7818
Description: Adding and subtracting one gives back the original set. Similar to pncan 9073 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 5899 . . . 4  |-  ( A  +c  1o )  e. 
_V
21a1i 10 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( A  +c  1o )  e. 
_V )
3 id 19 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  B  e.  ( A  +c  1o ) )
4 df1o2 6507 . . . . . . . 8  |-  1o  =  { (/) }
54xpeq1i 4725 . . . . . . 7  |-  ( 1o 
X.  { 1o }
)  =  ( {
(/) }  X.  { 1o } )
6 0ex 4166 . . . . . . . 8  |-  (/)  e.  _V
7 1on 6502 . . . . . . . . 9  |-  1o  e.  On
87elexi 2810 . . . . . . . 8  |-  1o  e.  _V
96, 8xpsn 5716 . . . . . . 7  |-  ( {
(/) }  X.  { 1o } )  =  { <.
(/) ,  1o >. }
105, 9eqtri 2316 . . . . . 6  |-  ( 1o 
X.  { 1o }
)  =  { <. (/)
,  1o >. }
11 ssun2 3352 . . . . . 6  |-  ( 1o 
X.  { 1o }
)  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
1210, 11eqsstr3i 3222 . . . . 5  |-  { <. (/)
,  1o >. }  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
13 opex 4253 . . . . . 6  |-  <. (/) ,  1o >.  e.  _V
1413snss 3761 . . . . 5  |-  ( <. (/)
,  1o >.  e.  ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  <->  { <. (/) ,  1o >. }  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
1512, 14mpbir 200 . . . 4  |-  <. (/) ,  1o >.  e.  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
16 relxp 4810 . . . . . . . 8  |-  Rel  ( _V  X.  _V )
17 cdafn 7811 . . . . . . . . . 10  |-  +c  Fn  ( _V  X.  _V )
18 fndm 5359 . . . . . . . . . 10  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
1917, 18ax-mp 8 . . . . . . . . 9  |-  dom  +c  =  ( _V  X.  _V )
2019releqi 4788 . . . . . . . 8  |-  ( Rel 
dom  +c  <->  Rel  ( _V  X.  _V ) )
2116, 20mpbir 200 . . . . . . 7  |-  Rel  dom  +c
2221ovrcl 5904 . . . . . 6  |-  ( B  e.  ( A  +c  1o )  ->  ( A  e.  _V  /\  1o  e.  _V ) )
2322simpld 445 . . . . 5  |-  ( B  e.  ( A  +c  1o )  ->  A  e. 
_V )
24 cdaval 7812 . . . . 5  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
2523, 7, 24sylancl 643 . . . 4  |-  ( B  e.  ( A  +c  1o )  ->  ( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
2615, 25syl5eleqr 2383 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
27 difsnen 6960 . . 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 1182 . 2  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { B }
)  ~~  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } ) )
2925difeq1d 3306 . . . 4  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )  \  { <. (/) ,  1o >. } ) )
30 xp01disj 6511 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
31 disj3 3512 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  i^i  ( 1o  X.  { 1o }
) )  =  (/)  <->  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) ) )
3230, 31mpbi 199 . . . . 5  |-  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) )
33 difun2 3546 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  \  ( 1o  X.  { 1o }
) )
3410difeq2i 3304 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )  \  { <. (/) ,  1o >. } )
3532, 33, 343eqtr2i 2322 . . . 4  |-  ( A  X.  { (/) } )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o 
X.  { 1o }
) )  \  { <.
(/) ,  1o >. } )
3629, 35syl6eqr 2346 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  =  ( A  X.  { (/) } ) )
37 xpsneng 6963 . . . 4  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
3823, 6, 37sylancl 643 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
3936, 38eqbrtrd 4059 . 2  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  ~~  A
)
40 entr 6929 . 2  |-  ( ( ( ( A  +c  1o )  \  { B } )  ~~  (
( A  +c  1o )  \  { <. (/) ,  1o >. } )  /\  (
( A  +c  1o )  \  { <. (/) ,  1o >. } )  ~~  A
)  ->  ( ( A  +c  1o )  \  { B } )  ~~  A )
4128, 39, 40syl2anc 642 1  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { B }
)  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039   Oncon0 4408    X. cxp 4703   dom cdm 4705   Rel wrel 4710    Fn wfn 5266  (class class class)co 5874   1oc1o 6488    ~~ cen 6876    +c ccda 7809
This theorem is referenced by:  canthp1  8292
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-3or 935  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-er 6676  df-en 6880  df-cda 7810
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