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Theorem omelon2 7864
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 7863 . . . 4 (ω ∈ On ∨ ω = On)
21ori 858 . . 3 (¬ ω ∈ On → ω = On)
3 onprc 7761 . . . 4 ¬ On ∈ V
4 eleq1 2815 . . . 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 3468  Oncon0 6357  ωcom 7851
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-lim 6362  df-om 7852
This theorem is referenced by:  oaabs  8646  omelon  9640  fictb  10239  axdc3lem  10444
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