Theorem omelon2 7858

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 7857	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
2	1	ori 861	. . 3 ⊢ (¬ ω ∈ On → ω = On)
3		onprc 7757	. . . 4 ⊢ ¬ On ∈ V
4		eleq1 2817	. . . 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 1540   ∈ wcel 2109  Vcvv 3450  Oncon0 6335  ωcom 7845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339  df-lim 6340  df-om 7846

This theorem is referenced by:  oaabs  8615  omelon  9606  fictb  10204  axdc3lem  10410