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Theorem infcda1 8075
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 7117 . . . . . . . 8  |-  Rel  ~<_
21brrelex2i 4921 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
3 1on 6733 . . . . . . 7  |-  1o  e.  On
4 cdaval 8052 . . . . . . 7  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
52, 3, 4sylancl 645 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  +c  1o )  =  ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) ) )
6 df1o2 6738 . . . . . . . . 9  |-  1o  =  { (/) }
76xpeq1i 4900 . . . . . . . 8  |-  ( 1o 
X.  { 1o }
)  =  ( {
(/) }  X.  { 1o } )
8 0ex 4341 . . . . . . . . 9  |-  (/)  e.  _V
93elexi 2967 . . . . . . . . 9  |-  1o  e.  _V
108, 9xpsn 5912 . . . . . . . 8  |-  ( {
(/) }  X.  { 1o } )  =  { <.
(/) ,  1o >. }
117, 10eqtr2i 2459 . . . . . . 7  |-  { <. (/)
,  1o >. }  =  ( 1o  X.  { 1o } )
1211a1i 11 . . . . . 6  |-  ( om  ~<_  A  ->  { <. (/) ,  1o >. }  =  ( 1o 
X.  { 1o }
) )
135, 12difeq12d 3468 . . . . 5  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  =  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) ) )
14 difun2 3709 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  \  ( 1o  X.  { 1o }
) )
15 xp01disj 6742 . . . . . . 7  |-  ( ( A  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
16 disj3 3674 . . . . . . 7  |-  ( ( ( A  X.  { (/)
} )  i^i  ( 1o  X.  { 1o }
) )  =  (/)  <->  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) ) )
1715, 16mpbi 201 . . . . . 6  |-  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) )
1814, 17eqtr4i 2461 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( A  X.  { (/) } )
1913, 18syl6eq 2486 . . . 4  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  =  ( A  X.  { (/) } ) )
20 cdadom3 8070 . . . . . . 7  |-  ( ( A  e.  _V  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
212, 3, 20sylancl 645 . . . . . 6  |-  ( om  ~<_  A  ->  A  ~<_  ( A  +c  1o ) )
22 domtr 7162 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  +c  1o ) )  ->  om  ~<_  ( A  +c  1o ) )
2321, 22mpdan 651 . . . . 5  |-  ( om  ~<_  A  ->  om  ~<_  ( A  +c  1o ) )
24 infdifsn 7613 . . . . 5  |-  ( om  ~<_  ( A  +c  1o )  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  ~~  ( A  +c  1o ) )
2523, 24syl 16 . . . 4  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  ~~  ( A  +c  1o ) )
2619, 25eqbrtrrd 4236 . . 3  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )
2726ensymd 7160 . 2  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  ( A  X.  { (/) } ) )
28 xpsneng 7195 . . 3  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
292, 8, 28sylancl 645 . 2  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  A )
30 entr 7161 . 2  |-  ( ( ( A  +c  1o )  ~~  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~~  A )  ->  ( A  +c  1o )  ~~  A )
3127, 29, 30syl2anc 644 1  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321   (/)c0 3630   {csn 3816   <.cop 3819   class class class wbr 4214   Oncon0 4583   omcom 4847    X. cxp 4878  (class class class)co 6083   1oc1o 6719    ~~ cen 7108    ~<_ cdom 7109    +c ccda 8049
This theorem is referenced by:  pwcdaidm  8077  isfin4-3  8197  canthp1lem2  8530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1o 6726  df-er 6907  df-en 7112  df-dom 7113  df-cda 8050
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