MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin2inf Structured version   Unicode version

Theorem fin2inf 7372
Description: This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless 
om exists. (Contributed by NM, 13-Nov-2003.)
Assertion
Ref Expression
fin2inf  |-  ( A 
~<  om  ->  om  e.  _V )

Proof of Theorem fin2inf
StepHypRef Expression
1 relsdom 7118 . 2  |-  Rel  ~<
21brrelex2i 4921 1  |-  ( A 
~<  om  ->  om  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   _Vcvv 2958   class class class wbr 4214   omcom 4847    ~< csdm 7110
This theorem is referenced by:  unfi2  7378  unifi2  7398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-dom 7113  df-sdom 7114
  Copyright terms: Public domain W3C validator