Theorem fin2inf 9280

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 8964	. 2 ⊢ Rel ≺
2	1	brrelex2i 5708	1 ⊢ (𝐴 ≺ ω → ω ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451   class class class wbr 5103  ωcom 7866   ≺ csdm 8956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dom 8959  df-sdom 8960

This theorem is used by:  unfi2  9286  unifi2  9318