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Theorem infcda1 7819
Description: An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
infcda1  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A
)

Proof of Theorem infcda1
StepHypRef Expression
1 reldom 6869 . . . . . . . 8  |-  Rel  ~<_
21brrelex2i 4730 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
3 1on 6486 . . . . . . 7  |-  1o  e.  On
4 cdaval 7796 . . . . . . 7  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
52, 3, 4sylancl 643 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  +c  1o )  =  ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) ) )
6 df1o2 6491 . . . . . . . . 9  |-  1o  =  { (/) }
76xpeq1i 4709 . . . . . . . 8  |-  ( 1o 
X.  { 1o }
)  =  ( {
(/) }  X.  { 1o } )
8 0ex 4150 . . . . . . . . 9  |-  (/)  e.  _V
93elexi 2797 . . . . . . . . 9  |-  1o  e.  _V
108, 9xpsn 5700 . . . . . . . 8  |-  ( {
(/) }  X.  { 1o } )  =  { <.
(/) ,  1o >. }
117, 10eqtr2i 2304 . . . . . . 7  |-  { <. (/)
,  1o >. }  =  ( 1o  X.  { 1o } )
1211a1i 10 . . . . . 6  |-  ( om  ~<_  A  ->  { <. (/) ,  1o >. }  =  ( 1o 
X.  { 1o }
) )
135, 12difeq12d 3295 . . . . 5  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  =  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) ) )
14 difun2 3533 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  \  ( 1o  X.  { 1o }
) )
15 xp01disj 6495 . . . . . . 7  |-  ( ( A  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
16 disj3 3499 . . . . . . 7  |-  ( ( ( A  X.  { (/)
} )  i^i  ( 1o  X.  { 1o }
) )  =  (/)  <->  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) ) )
1715, 16mpbi 199 . . . . . 6  |-  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) )
1814, 17eqtr4i 2306 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( A  X.  { (/) } )
1913, 18syl6eq 2331 . . . 4  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  =  ( A  X.  { (/) } ) )
20 cdadom3 7814 . . . . . . 7  |-  ( ( A  e.  _V  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
212, 3, 20sylancl 643 . . . . . 6  |-  ( om  ~<_  A  ->  A  ~<_  ( A  +c  1o ) )
22 domtr 6914 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  +c  1o ) )  ->  om  ~<_  ( A  +c  1o ) )
2321, 22mpdan 649 . . . . 5  |-  ( om  ~<_  A  ->  om  ~<_  ( A  +c  1o ) )
24 infdifsn 7357 . . . . 5  |-  ( om  ~<_  ( A  +c  1o )  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  ~~  ( A  +c  1o ) )
2523, 24syl 15 . . . 4  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  ~~  ( A  +c  1o ) )
2619, 25eqbrtrrd 4045 . . 3  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )
27 ensym 6910 . . 3  |-  ( ( A  X.  { (/) } )  ~~  ( A  +c  1o )  -> 
( A  +c  1o )  ~~  ( A  X.  { (/) } ) )
2826, 27syl 15 . 2  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  ( A  X.  { (/) } ) )
29 xpsneng 6947 . . 3  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
302, 8, 29sylancl 643 . 2  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  A )
31 entr 6913 . 2  |-  ( ( ( A  +c  1o )  ~~  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~~  A )  ->  ( A  +c  1o )  ~~  A )
3228, 30, 31syl2anc 642 1  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  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   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   Oncon0 4392   omcom 4656    X. cxp 4687  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    +c ccda 7793
This theorem is referenced by:  pwcdaidm  7821  isfin4-3  7941  canthp1lem2  8275
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-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-lim 4397  df-suc 4398  df-om 4657  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-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-cda 7794
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