Theorem fin2inf 9252

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 8938	. 2 ⊢ Rel ≺
2	1	brrelex2i 5708	1 ⊢ (𝐴 ≺ ω → ω ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3457   class class class wbr 5104  ωcom 7850   ≺ csdm 8930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5657  df-rel 5658  df-dom 8933  df-sdom 8934

This theorem is referenced by:  unfi2  9258  unifi2  9290