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Theorem cdainf 8028
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 7074 . . . . 5  |-  Rel  ~<_
21brrelex2i 4878 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
3 cdadom3 8024 . . . 4  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  A  ~<_  ( A  +c  A ) )
42, 2, 3syl2anc 643 . . 3  |-  ( om  ~<_  A  ->  A  ~<_  ( A  +c  A ) )
5 domtr 7119 . . 3  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  +c  A
) )  ->  om  ~<_  ( A  +c  A ) )
64, 5mpdan 650 . 2  |-  ( om  ~<_  A  ->  om  ~<_  ( A  +c  A ) )
7 infn0 7328 . . . 4  |-  ( om  ~<_  ( A  +c  A
)  ->  ( A  +c  A )  =/=  (/) )
8 cdafn 8005 . . . . . . . 8  |-  +c  Fn  ( _V  X.  _V )
9 fndm 5503 . . . . . . . 8  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
108, 9ax-mp 8 . . . . . . 7  |-  dom  +c  =  ( _V  X.  _V )
1110ndmov 6190 . . . . . 6  |-  ( -.  ( A  e.  _V  /\  A  e.  _V )  ->  ( A  +c  A
)  =  (/) )
1211necon1ai 2609 . . . . 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 6065 . . . . 5  |-  ( A  +c  A )  e. 
_V
1615domen 7080 . . . 4  |-  ( om  ~<_  ( A  +c  A
)  <->  E. x ( om 
~~  x  /\  x  C_  ( A  +c  A
) ) )
17 indi 3547 . . . . . . . . 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 8006 . . . . . . . . . . . 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 3344 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  x  C_  (
( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) ) )
23 df-ss 3294 . . . . . . . . . 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 2450 . . . . . . . 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 7115 . . . . . . . . 9  |-  ( om 
~~  x  ->  x  ~~  om )
2726ad2antrl 709 . . . . . . . 8  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  x  ~~  om )
2825, 27eqbrtrd 4192 . . . . . . 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 8027 . . . . . . 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 4365 . . . . . . . . . . . 12  |-  { (/) }  e.  _V
32 xpexg 4948 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
3331, 32mpan2 653 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  X.  { (/) } )  e.  _V )
34 inss2 3522 . . . . . . . . . . 11  |-  ( x  i^i  ( A  X.  { (/) } ) ) 
C_  ( A  X.  { (/) } )
35 ssdomg 7112 . . . . . . . . . . 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 4299 . . . . . . . . . . 11  |-  (/)  e.  _V
38 xpsneng 7152 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
3937, 38mpan2 653 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  { (/) } ) 
~~  A )
40 domentr 7125 . . . . . . . . . 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 7208 . . . . . . . . 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 4365 . . . . . . . . . . . 12  |-  { 1o }  e.  _V
45 xpexg 4948 . . . . . . . . . . . 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 3522 . . . . . . . . . . 11  |-  ( x  i^i  ( A  X.  { 1o } ) ) 
C_  ( A  X.  { 1o } )
48 ssdomg 7112 . . . . . . . . . . 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 6690 . . . . . . . . . . 11  |-  1o  e.  On
51 xpsneng 7152 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
5250, 51mpan2 653 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  ~~  A )
53 domentr 7125 . . . . . . . . . 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 7208 . . . . . . . . 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 2567   _Vcvv 2916    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172   Oncon0 4541   omcom 4804    X. cxp 4835   dom cdm 4837    Fn wfn 5408  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066    +c ccda 8003
This theorem is referenced by:  infdif  8045
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-iun 4055  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-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-cda 8004
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