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Theorem omon 7888
Description: The class of natural numbers ω is either an ordinal number (if we accept the Axiom of Infinity) or the proper class of all ordinal numbers (if we deny the Axiom of Infinity). Remark in [TakeutiZaring] p. 43. (Contributed by NM, 10-May-1998.)
Assertion
Ref Expression
omon (ω ∈ On ∨ ω = On)

Proof of Theorem omon
StepHypRef Expression
1 ordom 7886 . 2 Ord ω
2 ordeleqon 7790 . 2 (Ord ω ↔ (ω ∈ On ∨ ω = On))
31, 2mpbi 229 1 (ω ∈ On ∨ ω = On)
Colors of variables: wff setvar class
Syntax hints:  wo 845   = wceq 1534  wcel 2099  Ord word 6375  Oncon0 6376  ωcom 7876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380  df-lim 6381  df-om 7877
This theorem is referenced by:  omelon2  7889  infensuc  9193  elhf2  35999
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