Theorem omelon2 7823

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 7822	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
2	1	ori 862	. . 3 ⊢ (¬ ω ∈ On → ω = On)
3		onprc 7725	. . . 4 ⊢ ¬ On ∈ V
4		eleq1 2825	. . . 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 3441  Oncon0 6318  ωcom 7810

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 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-lim 6323  df-om 7811

This theorem is referenced by:  oaabs  8578  omelon  9559  fictb  10158  axdc3lem  10364  n0ssoldg  28333