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Theorem fin2inf 7136
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 6886 . 2  |-  Rel  ~<
21brrelex2i 4746 1  |-  ( A 
~<  om  ->  om  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   omcom 4672    ~< csdm 6878
This theorem is referenced by:  unfi2  7142  unifi2  7162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dom 6881  df-sdom 6882
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