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Theorem omon 7566
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 7564 . 2 Ord ω
2 ordeleqon 7478 . 2 (Ord ω ↔ (ω ∈ On ∨ ω = On))
31, 2mpbi 233 1 (ω ∈ On ∨ ω = On)
Colors of variables: wff setvar class
Syntax hints:  wo 844   = wceq 1538  wcel 2115  Ord word 6163  Oncon0 6164  ωcom 7555
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-tr 5146  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-om 7556
This theorem is referenced by:  omelon2  7567  infensuc  8671  elhf2  33643
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