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Theorem omelon2 7862
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
StepHypRef Expression
1 omon 7861 . . . 4 (ω ∈ On ∨ ω = On)
21ori 858 . . 3 (¬ ω ∈ On → ω = On)
3 onprc 7759 . . . 4 ¬ On ∈ V
4 eleq1 2813 . . . 4 (ω = On → (ω ∈ V ↔ On ∈ V))
53, 4mtbiri 327 . . 3 (ω = On → ¬ ω ∈ V)
62, 5syl 17 . 2 (¬ ω ∈ On → ¬ ω ∈ V)
76con4i 114 1 (ω ∈ V → ω ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  Vcvv 3466  Oncon0 6355  ωcom 7849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-lim 6360  df-om 7850
This theorem is referenced by:  oaabs  8644  omelon  9638  fictb  10237  axdc3lem  10442
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