Theorem omelon2 7873

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 7872	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
2	1	ori 875	. . 3 ⊢ (¬ ω ∈ On → ω = On)
3		onprc 7775	. . . 4 ⊢ ¬ On ∈ V
4		eleq1 2848	. . . 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 3450  Oncon0 6351  ωcom 7860

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 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6354  df-on 6355  df-lim 6356  df-om 7861

This theorem is used by:  oaabs  8635  omelon  9625  fictb  10293  axdc3lem  10499  n0ssoldg  28672