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Theorem cda1dif 8012
Description: Adding and subtracting one gives back the original set. Similar to pncan 9267 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 6065 . . . 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 6695 . . . . . . . 8  |-  1o  =  { (/) }
54xpeq1i 4857 . . . . . . 7  |-  ( 1o 
X.  { 1o }
)  =  ( {
(/) }  X.  { 1o } )
6 0ex 4299 . . . . . . . 8  |-  (/)  e.  _V
7 1on 6690 . . . . . . . . 9  |-  1o  e.  On
87elexi 2925 . . . . . . . 8  |-  1o  e.  _V
96, 8xpsn 5869 . . . . . . 7  |-  ( {
(/) }  X.  { 1o } )  =  { <.
(/) ,  1o >. }
105, 9eqtri 2424 . . . . . 6  |-  ( 1o 
X.  { 1o }
)  =  { <. (/)
,  1o >. }
11 ssun2 3471 . . . . . 6  |-  ( 1o 
X.  { 1o }
)  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
1210, 11eqsstr3i 3339 . . . . 5  |-  { <. (/)
,  1o >. }  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
13 opex 4387 . . . . . 6  |-  <. (/) ,  1o >.  e.  _V
1413snss 3886 . . . . 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 4942 . . . . . . . 8  |-  Rel  ( _V  X.  _V )
17 cdafn 8005 . . . . . . . . . 10  |-  +c  Fn  ( _V  X.  _V )
18 fndm 5503 . . . . . . . . . 10  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
1917, 18ax-mp 8 . . . . . . . . 9  |-  dom  +c  =  ( _V  X.  _V )
2019releqi 4919 . . . . . . . 8  |-  ( Rel 
dom  +c  <->  Rel  ( _V  X.  _V ) )
2116, 20mpbir 201 . . . . . . 7  |-  Rel  dom  +c
2221ovrcl 6070 . . . . . 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 8006 . . . . 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 2491 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
27 difsnen 7149 . . 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 3424 . . . 4  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )  \  { <. (/) ,  1o >. } ) )
30 xp01disj 6699 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
31 disj3 3632 . . . . . 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 3667 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  \  ( 1o  X.  { 1o }
) )
3410difeq2i 3422 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )  \  { <. (/) ,  1o >. } )
3532, 33, 343eqtr2i 2430 . . . 4  |-  ( A  X.  { (/) } )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o 
X.  { 1o }
) )  \  { <.
(/) ,  1o >. } )
3629, 35syl6eqr 2454 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  =  ( A  X.  { (/) } ) )
37 xpsneng 7152 . . . 4  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
3823, 6, 37sylancl 644 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
3936, 38eqbrtrd 4192 . 2  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  ~~  A
)
40 entr 7118 . 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 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172   Oncon0 4541    X. cxp 4835   dom cdm 4837   Rel wrel 4842    Fn wfn 5408  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    +c ccda 8003
This theorem is referenced by:  canthp1  8485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-1o 6683  df-er 6864  df-en 7069  df-cda 8004
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