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Theorem cdainf 8036
Description: A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cdainf  |-  ( om  ~<_  A  <->  om  ~<_  ( A  +c  A ) )

Proof of Theorem cdainf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 reldom 7082 . . . . 5  |-  Rel  ~<_
21brrelex2i 4886 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
3 cdadom3 8032 . . . 4  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  A  ~<_  ( A  +c  A ) )
42, 2, 3syl2anc 643 . . 3  |-  ( om  ~<_  A  ->  A  ~<_  ( A  +c  A ) )
5 domtr 7127 . . 3  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  +c  A
) )  ->  om  ~<_  ( A  +c  A ) )
64, 5mpdan 650 . 2  |-  ( om  ~<_  A  ->  om  ~<_  ( A  +c  A ) )
7 infn0 7336 . . . 4  |-  ( om  ~<_  ( A  +c  A
)  ->  ( A  +c  A )  =/=  (/) )
8 cdafn 8013 . . . . . . . 8  |-  +c  Fn  ( _V  X.  _V )
9 fndm 5511 . . . . . . . 8  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
108, 9ax-mp 8 . . . . . . 7  |-  dom  +c  =  ( _V  X.  _V )
1110ndmov 6198 . . . . . 6  |-  ( -.  ( A  e.  _V  /\  A  e.  _V )  ->  ( A  +c  A
)  =  (/) )
1211necon1ai 2617 . . . . 5  |-  ( ( A  +c  A )  =/=  (/)  ->  ( A  e.  _V  /\  A  e. 
_V ) )
1312simpld 446 . . . 4  |-  ( ( A  +c  A )  =/=  (/)  ->  A  e.  _V )
147, 13syl 16 . . 3  |-  ( om  ~<_  ( A  +c  A
)  ->  A  e.  _V )
15 ovex 6073 . . . . 5  |-  ( A  +c  A )  e. 
_V
1615domen 7088 . . . 4  |-  ( om  ~<_  ( A  +c  A
)  <->  E. x ( om 
~~  x  /\  x  C_  ( A  +c  A
) ) )
17 indi 3555 . . . . . . . . 9  |-  ( x  i^i  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) ) )  =  ( ( x  i^i  ( A  X.  { (/) } ) )  u.  ( x  i^i  ( A  X.  { 1o } ) ) )
18 simprr 734 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  x  C_  ( A  +c  A ) )
19 simpl 444 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  A  e.  _V )
20 cdaval 8014 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  +c  A
)  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
2119, 19, 20syl2anc 643 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  ( A  +c  A )  =  ( ( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) ) )
2218, 21sseqtrd 3352 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  x  C_  (
( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) ) )
23 df-ss 3302 . . . . . . . . . 10  |-  ( x 
C_  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) )  <-> 
( x  i^i  (
( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) ) )  =  x )
2422, 23sylib 189 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  ( x  i^i  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) ) )  =  x )
2517, 24syl5eqr 2458 . . . . . . . 8  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  ( ( x  i^i  ( A  X.  { (/) } ) )  u.  ( x  i^i  ( A  X.  { 1o } ) ) )  =  x )
26 ensym 7123 . . . . . . . . 9  |-  ( om 
~~  x  ->  x  ~~  om )
2726ad2antrl 709 . . . . . . . 8  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  x  ~~  om )
2825, 27eqbrtrd 4200 . . . . . . 7  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  ( ( x  i^i  ( A  X.  { (/) } ) )  u.  ( x  i^i  ( A  X.  { 1o } ) ) ) 
~~  om )
2928ex 424 . . . . . 6  |-  ( A  e.  _V  ->  (
( om  ~~  x  /\  x  C_  ( A  +c  A ) )  ->  ( ( x  i^i  ( A  X.  { (/) } ) )  u.  ( x  i^i  ( A  X.  { 1o } ) ) ) 
~~  om ) )
30 cdainflem 8035 . . . . . . 7  |-  ( ( ( x  i^i  ( A  X.  { (/) } ) )  u.  ( x  i^i  ( A  X.  { 1o } ) ) )  ~~  om  ->  ( ( x  i^i  ( A  X.  { (/) } ) )  ~~  om  \/  ( x  i^i  ( A  X.  { 1o }
) )  ~~  om ) )
31 snex 4373 . . . . . . . . . . . 12  |-  { (/) }  e.  _V
32 xpexg 4956 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
3331, 32mpan2 653 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  X.  { (/) } )  e.  _V )
34 inss2 3530 . . . . . . . . . . 11  |-  ( x  i^i  ( A  X.  { (/) } ) ) 
C_  ( A  X.  { (/) } )
35 ssdomg 7120 . . . . . . . . . . 11  |-  ( ( A  X.  { (/) } )  e.  _V  ->  ( ( x  i^i  ( A  X.  { (/) } ) )  C_  ( A  X.  { (/) } )  -> 
( x  i^i  ( A  X.  { (/) } ) )  ~<_  ( A  X.  { (/) } ) ) )
3633, 34, 35ee10 1382 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
x  i^i  ( A  X.  { (/) } ) )  ~<_  ( A  X.  { (/)
} ) )
37 0ex 4307 . . . . . . . . . . 11  |-  (/)  e.  _V
38 xpsneng 7160 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
3937, 38mpan2 653 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  { (/) } ) 
~~  A )
40 domentr 7133 . . . . . . . . . 10  |-  ( ( ( x  i^i  ( A  X.  { (/) } ) )  ~<_  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~~  A )  ->  ( x  i^i  ( A  X.  { (/)
} ) )  ~<_  A )
4136, 39, 40syl2anc 643 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
x  i^i  ( A  X.  { (/) } ) )  ~<_  A )
42 domen1 7216 . . . . . . . . 9  |-  ( ( x  i^i  ( A  X.  { (/) } ) )  ~~  om  ->  ( ( x  i^i  ( A  X.  { (/) } ) )  ~<_  A  <->  om  ~<_  A ) )
4341, 42syl5ibcom 212 . . . . . . . 8  |-  ( A  e.  _V  ->  (
( x  i^i  ( A  X.  { (/) } ) )  ~~  om  ->  om  ~<_  A ) )
44 snex 4373 . . . . . . . . . . . 12  |-  { 1o }  e.  _V
45 xpexg 4956 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  { 1o }  e.  _V )  ->  ( A  X.  { 1o } )  e. 
_V )
4644, 45mpan2 653 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  e.  _V )
47 inss2 3530 . . . . . . . . . . 11  |-  ( x  i^i  ( A  X.  { 1o } ) ) 
C_  ( A  X.  { 1o } )
48 ssdomg 7120 . . . . . . . . . . 11  |-  ( ( A  X.  { 1o } )  e.  _V  ->  ( ( x  i^i  ( A  X.  { 1o } ) )  C_  ( A  X.  { 1o } )  ->  (
x  i^i  ( A  X.  { 1o } ) )  ~<_  ( A  X.  { 1o } ) ) )
4946, 47, 48ee10 1382 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
x  i^i  ( A  X.  { 1o } ) )  ~<_  ( A  X.  { 1o } ) )
50 1on 6698 . . . . . . . . . . 11  |-  1o  e.  On
51 xpsneng 7160 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
5250, 51mpan2 653 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  ~~  A )
53 domentr 7133 . . . . . . . . . 10  |-  ( ( ( x  i^i  ( A  X.  { 1o }
) )  ~<_  ( A  X.  { 1o }
)  /\  ( A  X.  { 1o } ) 
~~  A )  -> 
( x  i^i  ( A  X.  { 1o }
) )  ~<_  A )
5449, 52, 53syl2anc 643 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
x  i^i  ( A  X.  { 1o } ) )  ~<_  A )
55 domen1 7216 . . . . . . . . 9  |-  ( ( x  i^i  ( A  X.  { 1o }
) )  ~~  om  ->  ( ( x  i^i  ( A  X.  { 1o } ) )  ~<_  A  <->  om 
~<_  A ) )
5654, 55syl5ibcom 212 . . . . . . . 8  |-  ( A  e.  _V  ->  (
( x  i^i  ( A  X.  { 1o }
) )  ~~  om  ->  om  ~<_  A ) )
5743, 56jaod 370 . . . . . . 7  |-  ( A  e.  _V  ->  (
( ( x  i^i  ( A  X.  { (/)
} ) )  ~~  om  \/  ( x  i^i  ( A  X.  { 1o } ) )  ~~  om )  ->  om  ~<_  A ) )
5830, 57syl5 30 . . . . . 6  |-  ( A  e.  _V  ->  (
( ( x  i^i  ( A  X.  { (/)
} ) )  u.  ( x  i^i  ( A  X.  { 1o }
) ) )  ~~  om 
->  om  ~<_  A ) )
5929, 58syld 42 . . . . 5  |-  ( A  e.  _V  ->  (
( om  ~~  x  /\  x  C_  ( A  +c  A ) )  ->  om  ~<_  A )
)
6059exlimdv 1643 . . . 4  |-  ( A  e.  _V  ->  ( E. x ( om  ~~  x  /\  x  C_  ( A  +c  A ) )  ->  om  ~<_  A )
)
6116, 60syl5bi 209 . . 3  |-  ( A  e.  _V  ->  ( om 
~<_  ( A  +c  A
)  ->  om  ~<_  A ) )
6214, 61mpcom 34 . 2  |-  ( om  ~<_  ( A  +c  A
)  ->  om  ~<_  A )
636, 62impbii 181 1  |-  ( om  ~<_  A  <->  om  ~<_  ( A  +c  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2575   _Vcvv 2924    u. cun 3286    i^i cin 3287    C_ wss 3288   (/)c0 3596   {csn 3782   class class class wbr 4180   Oncon0 4549   omcom 4812    X. cxp 4843   dom cdm 4845    Fn wfn 5416  (class class class)co 6048   1oc1o 6684    ~~ cen 7073    ~<_ cdom 7074    +c ccda 8011
This theorem is referenced by:  infdif  8053
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-cda 8012
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