Theorem omelon2 7859

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 7858	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
2	1	ori 872	. . 3 ⊢ (¬ ω ∈ On → ω = On)
3		onprc 7761	. . . 4 ⊢ ¬ On ∈ V
4		eleq1 2851	. . . 4 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V))
5	3, 4	mtbiri 329	. . 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 1561   ∈ wcel 2143  Vcvv 3455  Oncon0 6346  ωcom 7846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-tr 5209  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-ord 6349  df-on 6350  df-lim 6351  df-om 7847

This theorem is referenced by:  oaabs  8618  omelon  9599  fictb  10211  axdc3lem  10418  n0ssoldg  28453