MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdainf Unicode version

Theorem cdainf 7818
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 6869 . . . . 5  |-  Rel  ~<_
21brrelex2i 4730 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
3 cdadom3 7814 . . . 4  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  A  ~<_  ( A  +c  A ) )
42, 2, 3syl2anc 642 . . 3  |-  ( om  ~<_  A  ->  A  ~<_  ( A  +c  A ) )
5 domtr 6914 . . 3  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  +c  A
) )  ->  om  ~<_  ( A  +c  A ) )
64, 5mpdan 649 . 2  |-  ( om  ~<_  A  ->  om  ~<_  ( A  +c  A ) )
7 infn0 7119 . . . 4  |-  ( om  ~<_  ( A  +c  A
)  ->  ( A  +c  A )  =/=  (/) )
8 cdafn 7795 . . . . . . . 8  |-  +c  Fn  ( _V  X.  _V )
9 fndm 5343 . . . . . . . 8  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
108, 9ax-mp 8 . . . . . . 7  |-  dom  +c  =  ( _V  X.  _V )
1110ndmov 6004 . . . . . 6  |-  ( -.  ( A  e.  _V  /\  A  e.  _V )  ->  ( A  +c  A
)  =  (/) )
1211necon1ai 2488 . . . . 5  |-  ( ( A  +c  A )  =/=  (/)  ->  ( A  e.  _V  /\  A  e. 
_V ) )
1312simpld 445 . . . 4  |-  ( ( A  +c  A )  =/=  (/)  ->  A  e.  _V )
147, 13syl 15 . . 3  |-  ( om  ~<_  ( A  +c  A
)  ->  A  e.  _V )
15 ovex 5883 . . . . 5  |-  ( A  +c  A )  e. 
_V
1615domen 6875 . . . 4  |-  ( om  ~<_  ( A  +c  A
)  <->  E. x ( om 
~~  x  /\  x  C_  ( A  +c  A
) ) )
17 indi 3415 . . . . . . . . 9  |-  ( x  i^i  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) ) )  =  ( ( x  i^i  ( A  X.  { (/) } ) )  u.  ( x  i^i  ( A  X.  { 1o } ) ) )
18 simprr 733 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  x  C_  ( A  +c  A ) )
19 simpl 443 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  A  e.  _V )
20 cdaval 7796 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  +c  A
)  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
2119, 19, 20syl2anc 642 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  ( A  +c  A )  =  ( ( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) ) )
2218, 21sseqtrd 3214 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  x  C_  (
( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) ) )
23 df-ss 3166 . . . . . . . . . 10  |-  ( x 
C_  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) )  <-> 
( x  i^i  (
( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) ) )  =  x )
2422, 23sylib 188 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  ( x  i^i  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) ) )  =  x )
2517, 24syl5eqr 2329 . . . . . . . 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 6910 . . . . . . . . 9  |-  ( om 
~~  x  ->  x  ~~  om )
2726ad2antrl 708 . . . . . . . 8  |-  ( ( A  e.  _V  /\  ( om  ~~  x  /\  x  C_  ( A  +c  A ) ) )  ->  x  ~~  om )
2825, 27eqbrtrd 4043 . . . . . . 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 423 . . . . . 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 7817 . . . . . . 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 4216 . . . . . . . . . . . 12  |-  { (/) }  e.  _V
32 xpexg 4800 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
3331, 32mpan2 652 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  X.  { (/) } )  e.  _V )
34 inss2 3390 . . . . . . . . . . 11  |-  ( x  i^i  ( A  X.  { (/) } ) ) 
C_  ( A  X.  { (/) } )
35 ssdomg 6907 . . . . . . . . . . 11  |-  ( ( A  X.  { (/) } )  e.  _V  ->  ( ( x  i^i  ( A  X.  { (/) } ) )  C_  ( A  X.  { (/) } )  -> 
( x  i^i  ( A  X.  { (/) } ) )  ~<_  ( A  X.  { (/) } ) ) )
3633, 34, 35ee10 1366 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
x  i^i  ( A  X.  { (/) } ) )  ~<_  ( A  X.  { (/)
} ) )
37 0ex 4150 . . . . . . . . . . 11  |-  (/)  e.  _V
38 xpsneng 6947 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
3937, 38mpan2 652 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  { (/) } ) 
~~  A )
40 domentr 6920 . . . . . . . . . 10  |-  ( ( ( x  i^i  ( A  X.  { (/) } ) )  ~<_  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~~  A )  ->  ( x  i^i  ( A  X.  { (/)
} ) )  ~<_  A )
4136, 39, 40syl2anc 642 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
x  i^i  ( A  X.  { (/) } ) )  ~<_  A )
42 domen1 7003 . . . . . . . . 9  |-  ( ( x  i^i  ( A  X.  { (/) } ) )  ~~  om  ->  ( ( x  i^i  ( A  X.  { (/) } ) )  ~<_  A  <->  om  ~<_  A ) )
4341, 42syl5ibcom 211 . . . . . . . 8  |-  ( A  e.  _V  ->  (
( x  i^i  ( A  X.  { (/) } ) )  ~~  om  ->  om  ~<_  A ) )
44 snex 4216 . . . . . . . . . . . 12  |-  { 1o }  e.  _V
45 xpexg 4800 . . . . . . . . . . . 12  |-  ( ( A  e.  _V  /\  { 1o }  e.  _V )  ->  ( A  X.  { 1o } )  e. 
_V )
4644, 45mpan2 652 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  e.  _V )
47 inss2 3390 . . . . . . . . . . 11  |-  ( x  i^i  ( A  X.  { 1o } ) ) 
C_  ( A  X.  { 1o } )
48 ssdomg 6907 . . . . . . . . . . 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 1366 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
x  i^i  ( A  X.  { 1o } ) )  ~<_  ( A  X.  { 1o } ) )
50 1on 6486 . . . . . . . . . . 11  |-  1o  e.  On
51 xpsneng 6947 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
5250, 51mpan2 652 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  ~~  A )
53 domentr 6920 . . . . . . . . . 10  |-  ( ( ( x  i^i  ( A  X.  { 1o }
) )  ~<_  ( A  X.  { 1o }
)  /\  ( A  X.  { 1o } ) 
~~  A )  -> 
( x  i^i  ( A  X.  { 1o }
) )  ~<_  A )
5449, 52, 53syl2anc 642 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
x  i^i  ( A  X.  { 1o } ) )  ~<_  A )
55 domen1 7003 . . . . . . . . 9  |-  ( ( x  i^i  ( A  X.  { 1o }
) )  ~~  om  ->  ( ( x  i^i  ( A  X.  { 1o } ) )  ~<_  A  <->  om 
~<_  A ) )
5654, 55syl5ibcom 211 . . . . . . . 8  |-  ( A  e.  _V  ->  (
( x  i^i  ( A  X.  { 1o }
) )  ~~  om  ->  om  ~<_  A ) )
5743, 56jaod 369 . . . . . . 7  |-  ( A  e.  _V  ->  (
( ( x  i^i  ( A  X.  { (/)
} ) )  ~~  om  \/  ( x  i^i  ( A  X.  { 1o } ) )  ~~  om )  ->  om  ~<_  A ) )
5830, 57syl5 28 . . . . . 6  |-  ( A  e.  _V  ->  (
( ( x  i^i  ( A  X.  { (/)
} ) )  u.  ( x  i^i  ( A  X.  { 1o }
) ) )  ~~  om 
->  om  ~<_  A ) )
5929, 58syld 40 . . . . 5  |-  ( A  e.  _V  ->  (
( om  ~~  x  /\  x  C_  ( A  +c  A ) )  ->  om  ~<_  A )
)
6059exlimdv 1664 . . . 4  |-  ( A  e.  _V  ->  ( E. x ( om  ~~  x  /\  x  C_  ( A  +c  A ) )  ->  om  ~<_  A )
)
6116, 60syl5bi 208 . . 3  |-  ( A  e.  _V  ->  ( om 
~<_  ( A  +c  A
)  ->  om  ~<_  A ) )
6214, 61mpcom 32 . 2  |-  ( om  ~<_  ( A  +c  A
)  ->  om  ~<_  A )
636, 62impbii 180 1  |-  ( om  ~<_  A  <->  om  ~<_  ( A  +c  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   Oncon0 4392   omcom 4656    X. cxp 4687   dom cdm 4689    Fn wfn 5250  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    +c ccda 7793
This theorem is referenced by:  infdif  7835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-cda 7794
  Copyright terms: Public domain W3C validator