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Theorem omelon2 7596
 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 7595 . . . 4 (ω ∈ On ∨ ω = On)
21ori 858 . . 3 (¬ ω ∈ On → ω = On)
3 onprc 7503 . . . 4 ¬ On ∈ V
4 eleq1 2839 . . . 4 (ω = On → (ω ∈ V ↔ On ∈ V))
53, 4mtbiri 330 . . 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 1538   ∈ wcel 2111  Vcvv 3409  Oncon0 6173  ωcom 7584 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-tr 5142  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-ord 6176  df-on 6177  df-lim 6178  df-om 7585 This theorem is referenced by:  oaabs  8286  omelon  9147  fictb  9710  axdc3lem  9915
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