Theorem omon 7822

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

Step	Hyp	Ref	Expression
1		ordom 7820	. 2 ⊢ Ord ω
2		ordeleqon 7729	. 2 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3	1, 2	mpbi 232	1 ⊢ (ω ∈ On ∨ ω = On)

Syntax hints:   ∨ wo 854   = wceq 1548   ∈ wcel 2121  Ord word 6313  Oncon0 6314  ωcom 7810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-tr 5183  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-lim 6319  df-om 7811

This theorem is referenced by:  omelon2  7823  infensuc  9087  xoromon  35285  elhf2  36418