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Theorem omelon2 7025
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 7024 . . . 4 (ω ∈ On ∨ ω = On)
21ori 390 . . 3 (¬ ω ∈ On → ω = On)
3 onprc 6932 . . . 4 ¬ On ∈ V
4 eleq1 2692 . . . 4 (ω = On → (ω ∈ V ↔ On ∈ V))
53, 4mtbiri 317 . . 3 (ω = On → ¬ ω ∈ V)
62, 5syl 17 . 2 (¬ ω ∈ On → ¬ ω ∈ V)
76con4i 113 1 (ω ∈ V → ω ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1992  Vcvv 3191  Oncon0 5685  ωcom 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-om 7014
This theorem is referenced by:  oaabs  7670  omelon  8488  fictb  9012  axdc3lem  9217
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