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Theorem fin2inf 9260
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 8897 . 2 Rel ≺
21brrelex2i 5694 1 (𝐴 ≺ ω → ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3448   class class class wbr 5110  ωcom 7807  csdm 8889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-dom 8892  df-sdom 8893
This theorem is referenced by:  unfi2  9266  unifi2  9293
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