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Theorem infpwfidom 6727
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 4003 . . 3  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
2 snfig 6461 . . 3  |-  ( x  e.  A  ->  { x }  e.  Fin )
31, 2elind 3169 . 2  |-  ( x  e.  A  ->  { x }  e.  ( ~P A  i^i  Fin ) )
4 sneqbg 3581 . . 3  |-  ( x  e.  A  ->  ( { x }  =  { y }  <->  x  =  y ) )
54adantr 270 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( { x }  =  { y }  <->  x  =  y ) )
63, 5dom2 6422 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 103    = wceq 1285    e. wcel 1434   _Vcvv 2612    i^i cin 2983   ~Pcpw 3406   {csn 3422   class class class wbr 3811    ~<_ cdom 6386   Fincfn 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1o 6113  df-en 6388  df-dom 6389  df-fin 6390
This theorem is referenced by: (None)
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