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Theorem omelon2 7899
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 7898 . . . 4 (ω ∈ On ∨ ω = On)
21ori 861 . . 3 (¬ ω ∈ On → ω = On)
3 onprc 7796 . . . 4 ¬ On ∈ V
4 eleq1 2826 . . . 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 1536  wcel 2105  Vcvv 3477  Oncon0 6385  ωcom 7886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-om 7887
This theorem is referenced by:  oaabs  8684  omelon  9683  fictb  10281  axdc3lem  10487
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