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Theorem fin2inf 8220
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 7959 . 2 Rel ≺
21brrelex2i 5157 1 (𝐴 ≺ ω → ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1989  Vcvv 3198   class class class wbr 4651  ωcom 7062  csdm 7951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-dom 7954  df-sdom 7955
This theorem is referenced by:  unfi2  8226  unifi2  8253
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