Theorem fin2inf 9214

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 8900	. 2 ⊢ Rel ≺
2	1	brrelex2i 5688	1 ⊢ (𝐴 ≺ ω → ω ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3429   class class class wbr 5085  ωcom 7817   ≺ csdm 8892

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dom 8895  df-sdom 8896

This theorem is referenced by:  unfi2  9220  unifi2  9255