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Theorem infpwfidom 7655
Description: The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption 
( ~P A  i^i  Fin )  e.  _V because this theorem also implies that  A is a set if  ~P A  i^i  Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
infpwfidom  |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )

Proof of Theorem infpwfidom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snelpwi 4220 . . 3  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
2 snfi 6941 . . . 4  |-  { x }  e.  Fin
32a1i 10 . . 3  |-  ( x  e.  A  ->  { x }  e.  Fin )
4 elin 3358 . . 3  |-  ( { x }  e.  ( ~P A  i^i  Fin ) 
<->  ( { x }  e.  ~P A  /\  {
x }  e.  Fin ) )
51, 3, 4sylanbrc 645 . 2  |-  ( x  e.  A  ->  { x }  e.  ( ~P A  i^i  Fin ) )
6 sneqbg 3783 . . 3  |-  ( x  e.  A  ->  ( { x }  =  { y }  <->  x  =  y ) )
76adantr 451 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( { x }  =  { y }  <->  x  =  y ) )
85, 7dom2 6904 1  |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   ~Pcpw 3625   {csn 3640   class class class wbr 4023    ~<_ cdom 6861   Fincfn 6863
This theorem is referenced by:  infpwfien  7689  ttukeylem1  8136  canthnum  8271
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-rep 4131  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-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-1o 6479  df-en 6864  df-dom 6865  df-fin 6867
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