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Theorem infcda1 7835
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 6885 . . . . . . . 8  |-  Rel  ~<_
21brrelex2i 4746 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
3 1on 6502 . . . . . . 7  |-  1o  e.  On
4 cdaval 7812 . . . . . . 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 6507 . . . . . . . . 9  |-  1o  =  { (/) }
76xpeq1i 4725 . . . . . . . 8  |-  ( 1o 
X.  { 1o }
)  =  ( {
(/) }  X.  { 1o } )
8 0ex 4166 . . . . . . . . 9  |-  (/)  e.  _V
93elexi 2810 . . . . . . . . 9  |-  1o  e.  _V
108, 9xpsn 5716 . . . . . . . 8  |-  ( {
(/) }  X.  { 1o } )  =  { <.
(/) ,  1o >. }
117, 10eqtr2i 2317 . . . . . . 7  |-  { <. (/)
,  1o >. }  =  ( 1o  X.  { 1o } )
1211a1i 10 . . . . . 6  |-  ( om  ~<_  A  ->  { <. (/) ,  1o >. }  =  ( 1o 
X.  { 1o }
) )
135, 12difeq12d 3308 . . . . 5  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  =  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) ) )
14 difun2 3546 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  \  ( 1o  X.  { 1o }
) )
15 xp01disj 6511 . . . . . . 7  |-  ( ( A  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
16 disj3 3512 . . . . . . 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 2319 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( A  X.  { (/) } )
1913, 18syl6eq 2344 . . . 4  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  =  ( A  X.  { (/) } ) )
20 cdadom3 7830 . . . . . . 7  |-  ( ( A  e.  _V  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
212, 3, 20sylancl 643 . . . . . 6  |-  ( om  ~<_  A  ->  A  ~<_  ( A  +c  1o ) )
22 domtr 6930 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  +c  1o ) )  ->  om  ~<_  ( A  +c  1o ) )
2321, 22mpdan 649 . . . . 5  |-  ( om  ~<_  A  ->  om  ~<_  ( A  +c  1o ) )
24 infdifsn 7373 . . . . 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 4061 . . 3  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )
27 ensym 6926 . . 3  |-  ( ( A  X.  { (/) } )  ~~  ( A  +c  1o )  -> 
( A  +c  1o )  ~~  ( A  X.  { (/) } ) )
2826, 27syl 15 . 2  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  ( A  X.  { (/) } ) )
29 xpsneng 6963 . . 3  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
302, 8, 29sylancl 643 . 2  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  A )
31 entr 6929 . 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 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039   Oncon0 4408   omcom 4672    X. cxp 4703  (class class class)co 5874   1oc1o 6488    ~~ cen 6876    ~<_ cdom 6877    +c ccda 7809
This theorem is referenced by:  pwcdaidm  7837  isfin4-3  7957  canthp1lem2  8291
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-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-lim 4413  df-suc 4414  df-om 4673  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-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-cda 7810
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