Theorem omelon2 7878

Description: Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)

Assertion

Ref	Expression
omelon2	⊢ (ω ∈ V → ω ∈ On)

Proof of Theorem omelon2

Step	Hyp	Ref	Expression
1		omon 7877	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
2	1	ori 875	. . 3 ⊢ (¬ ω ∈ On → ω = On)
3		onprc 7780	. . . 4 ⊢ ¬ On ∈ V
4		eleq1 2850	. . . 4 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V))
5	3, 4	mtbiri 330	. . 3 ⊢ (ω = On → ¬ ω ∈ V)
6	2, 5	syl 18	. 2 ⊢ (¬ ω ∈ On → ¬ ω ∈ V)
7	6	con4i 115	1 ⊢ (ω ∈ V → ω ∈ On)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  Oncon0 6361  ωcom 7865

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 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365  df-lim 6366  df-om 7866

This theorem is used by:  oaabs  8639  omelon  9628  fictb  10249  axdc3lem  10455  n0ssoldg  28616