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Theorem infcda1 8029
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 7074 . . . . . . . 8  |-  Rel  ~<_
21brrelex2i 4878 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
3 1on 6690 . . . . . . 7  |-  1o  e.  On
4 cdaval 8006 . . . . . . 7  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
52, 3, 4sylancl 644 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  +c  1o )  =  ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) ) )
6 df1o2 6695 . . . . . . . . 9  |-  1o  =  { (/) }
76xpeq1i 4857 . . . . . . . 8  |-  ( 1o 
X.  { 1o }
)  =  ( {
(/) }  X.  { 1o } )
8 0ex 4299 . . . . . . . . 9  |-  (/)  e.  _V
93elexi 2925 . . . . . . . . 9  |-  1o  e.  _V
108, 9xpsn 5869 . . . . . . . 8  |-  ( {
(/) }  X.  { 1o } )  =  { <.
(/) ,  1o >. }
117, 10eqtr2i 2425 . . . . . . 7  |-  { <. (/)
,  1o >. }  =  ( 1o  X.  { 1o } )
1211a1i 11 . . . . . 6  |-  ( om  ~<_  A  ->  { <. (/) ,  1o >. }  =  ( 1o 
X.  { 1o }
) )
135, 12difeq12d 3426 . . . . 5  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  =  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) ) )
14 difun2 3667 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  \  ( 1o  X.  { 1o }
) )
15 xp01disj 6699 . . . . . . 7  |-  ( ( A  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
16 disj3 3632 . . . . . . 7  |-  ( ( ( A  X.  { (/)
} )  i^i  ( 1o  X.  { 1o }
) )  =  (/)  <->  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) ) )
1715, 16mpbi 200 . . . . . 6  |-  ( A  X.  { (/) } )  =  ( ( A  X.  { (/) } ) 
\  ( 1o  X.  { 1o } ) )
1814, 17eqtr4i 2427 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  \  ( 1o  X.  { 1o }
) )  =  ( A  X.  { (/) } )
1913, 18syl6eq 2452 . . . 4  |-  ( om  ~<_  A  ->  ( ( A  +c  1o )  \  { <. (/) ,  1o >. } )  =  ( A  X.  { (/) } ) )
20 cdadom3 8024 . . . . . . 7  |-  ( ( A  e.  _V  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
212, 3, 20sylancl 644 . . . . . 6  |-  ( om  ~<_  A  ->  A  ~<_  ( A  +c  1o ) )
22 domtr 7119 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  +c  1o ) )  ->  om  ~<_  ( A  +c  1o ) )
2321, 22mpdan 650 . . . . 5  |-  ( om  ~<_  A  ->  om  ~<_  ( A  +c  1o ) )
24 infdifsn 7567 . . . . 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 4194 . . 3  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )
2726ensymd 7117 . 2  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  ( A  X.  { (/) } ) )
28 xpsneng 7152 . . 3  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
292, 8, 28sylancl 644 . 2  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  A )
30 entr 7118 . 2  |-  ( ( ( A  +c  1o )  ~~  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~~  A )  ->  ( A  +c  1o )  ~~  A )
3127, 29, 30syl2anc 643 1  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  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   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172   Oncon0 4541   omcom 4804    X. cxp 4835  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066    +c ccda 8003
This theorem is referenced by:  pwcdaidm  8031  isfin4-3  8151  canthp1lem2  8484
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-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-lim 4546  df-suc 4547  df-om 4805  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-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-cda 8004
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