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Theorem infpwfidom 7047
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 4129 . . 3  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
2 snfig 6701 . . 3  |-  ( x  e.  A  ->  { x }  e.  Fin )
31, 2elind 3256 . 2  |-  ( x  e.  A  ->  { x }  e.  ( ~P A  i^i  Fin ) )
4 sneqbg 3685 . . 3  |-  ( x  e.  A  ->  ( { x }  =  { y }  <->  x  =  y ) )
54adantr 274 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( { x }  =  { y }  <->  x  =  y ) )
63, 5dom2 6662 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 104    = wceq 1331    e. wcel 1480   _Vcvv 2681    i^i cin 3065   ~Pcpw 3505   {csn 3522   class class class wbr 3924    ~<_ cdom 6626   Fincfn 6627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1o 6306  df-en 6628  df-dom 6629  df-fin 6630
This theorem is referenced by: (None)
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