Theorem omelon2 7874

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 7873	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
2	1	ori 874	. . 3 ⊢ (¬ ω ∈ On → ω = On)
3		onprc 7776	. . . 4 ⊢ ¬ On ∈ V
4		eleq1 2849	. . . 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)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1568   ∈ wcel 2141  Vcvv 3453  Oncon0 6360  ωcom 7861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-lim 6365  df-om 7862

This theorem is referenced by:  oaabs  8633  omelon  9614  fictb  10226  axdc3lem  10433  n0ssoldg  28522