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Theorem cda1dif 7802
Description: Adding and subtracting one gives back the original set. Similar to pncan 9057 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 5883 . . . 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 6491 . . . . . . . 8  |-  1o  =  { (/) }
54xpeq1i 4709 . . . . . . 7  |-  ( 1o 
X.  { 1o }
)  =  ( {
(/) }  X.  { 1o } )
6 0ex 4150 . . . . . . . 8  |-  (/)  e.  _V
7 1on 6486 . . . . . . . . 9  |-  1o  e.  On
87elexi 2797 . . . . . . . 8  |-  1o  e.  _V
96, 8xpsn 5700 . . . . . . 7  |-  ( {
(/) }  X.  { 1o } )  =  { <.
(/) ,  1o >. }
105, 9eqtri 2303 . . . . . 6  |-  ( 1o 
X.  { 1o }
)  =  { <. (/)
,  1o >. }
11 ssun2 3339 . . . . . 6  |-  ( 1o 
X.  { 1o }
)  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
1210, 11eqsstr3i 3209 . . . . 5  |-  { <. (/)
,  1o >. }  C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
13 opex 4237 . . . . . 6  |-  <. (/) ,  1o >.  e.  _V
1413snss 3748 . . . . 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 4794 . . . . . . . 8  |-  Rel  ( _V  X.  _V )
17 cdafn 7795 . . . . . . . . . 10  |-  +c  Fn  ( _V  X.  _V )
18 fndm 5343 . . . . . . . . . 10  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
1917, 18ax-mp 8 . . . . . . . . 9  |-  dom  +c  =  ( _V  X.  _V )
2019releqi 4772 . . . . . . . 8  |-  ( Rel 
dom  +c  <->  Rel  ( _V  X.  _V ) )
2116, 20mpbir 200 . . . . . . 7  |-  Rel  dom  +c
2221ovrcl 5888 . . . . . 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 7796 . . . . 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 2370 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
27 difsnen 6944 . . 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 3293 . . . 4  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )  \  { <. (/) ,  1o >. } ) )
30 xp01disj 6495 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
31 disj3 3499 . . . . . 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 3533 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  \  ( 1o  X.  { 1o }
) )
3410difeq2i 3291 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )  \  { <. (/) ,  1o >. } )
3532, 33, 343eqtr2i 2309 . . . 4  |-  ( A  X.  { (/) } )  =  ( ( ( A  X.  { (/) } )  u.  ( 1o 
X.  { 1o }
) )  \  { <.
(/) ,  1o >. } )
3629, 35syl6eqr 2333 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  =  ( A  X.  { (/) } ) )
37 xpsneng 6947 . . . 4  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
3823, 6, 37sylancl 643 . . 3  |-  ( B  e.  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
3936, 38eqbrtrd 4043 . 2  |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { <. (/) ,  1o >. } )  ~~  A
)
40 entr 6913 . 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   Oncon0 4392    X. cxp 4687   dom cdm 4689   Rel wrel 4694    Fn wfn 5250  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    +c ccda 7793
This theorem is referenced by:  canthp1  8276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-er 6660  df-en 6864  df-cda 7794
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