Theorem omelon2 7819

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 7818	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
2	1	ori 862	. . 3 ⊢ (¬ ω ∈ On → ω = On)
3		onprc 7721	. . . 4 ⊢ ¬ On ∈ V
4		eleq1 2823	. . . 4 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V))
5	3, 4	mtbiri 327	. . 3 ⊢ (ω = On → ¬ ω ∈ V)
6	2, 5	syl 17	. 2 ⊢ (¬ ω ∈ On → ¬ ω ∈ V)
7	6	con4i 114	1 ⊢ (ω ∈ V → ω ∈ On)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3427  Oncon0 6312  ωcom 7806

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 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-tr 5182  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-ord 6315  df-on 6316  df-lim 6317  df-om 7807

This theorem is referenced by:  oaabs  8573  omelon  9556  fictb  10155  axdc3lem  10361  n0ssoldg  28333