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Theorem fin2inf 8778
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 ω exists. (Contributed by NM, 13-Nov-2003.)
Assertion
Ref Expression
fin2inf (𝐴 ≺ ω → ω ∈ V)

Proof of Theorem fin2inf
StepHypRef Expression
1 relsdom 8512 . 2 Rel ≺
21brrelex2i 5596 1 (𝐴 ≺ ω → ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  Vcvv 3480   class class class wbr 5052  ωcom 7574  csdm 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-dom 8507  df-sdom 8508
This theorem is referenced by:  unfi2  8784  unifi2  8811
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