Theorem fin2inf 9211

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 8886	. 2 ⊢ Rel ≺
2	1	brrelex2i 5680	1 ⊢ (𝐴 ≺ ω → ω ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3438   class class class wbr 5095  ωcom 7806   ≺ csdm 8878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-dom 8881  df-sdom 8882

This theorem is referenced by:  unfi2  9217  unifi2  9254