Theorem fin2inf 9324

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

Step	Hyp	Ref	Expression
1		relsdom 8974	. 2 ⊢ Rel ≺
2	1	brrelex2i 5722	1 ⊢ (𝐴 ≺ ω → ω ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3463   class class class wbr 5123  ωcom 7869   ≺ csdm 8966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-dom 8969  df-sdom 8970

This theorem is referenced by:  unfi2  9330  unifi2  9367