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Theorem isfiniteg 7113
Description: A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
isfiniteg  |-  ( om  e.  _V  ->  ( A  e.  Fin  <->  A  ~<  om ) )

Proof of Theorem isfiniteg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfi 6881 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
2 nnsdomg 7112 . . . . 5  |-  ( ( om  e.  _V  /\  x  e.  om )  ->  x  ~<  om )
3 sdomen1 7001 . . . . 5  |-  ( A 
~~  x  ->  ( A  ~<  om  <->  x  ~<  om )
)
42, 3syl5ibrcom 213 . . . 4  |-  ( ( om  e.  _V  /\  x  e.  om )  ->  ( A  ~~  x  ->  A  ~<  om )
)
54rexlimdva 2668 . . 3  |-  ( om  e.  _V  ->  ( E. x  e.  om  A  ~~  x  ->  A  ~<  om ) )
61, 5syl5bi 208 . 2  |-  ( om  e.  _V  ->  ( A  e.  Fin  ->  A  ~<  om ) )
7 isfinite2 7111 . 2  |-  ( A 
~<  om  ->  A  e.  Fin )
86, 7impbid1 194 1  |-  ( om  e.  _V  ->  ( A  e.  Fin  <->  A  ~<  om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1685   E.wrex 2545   _Vcvv 2789   class class class wbr 4024   omcom 4655    ~~ cen 6856    ~< csdm 6858   Fincfn 6859
This theorem is referenced by:  unfi2  7122  unifi2  7142  isfinite  7349  axcclem  8079
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863
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