Theorem omon 7861

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 7859	. 2 ⊢ Ord ω
2		ordeleqon 7763	. 2 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3	1, 2	mpbi 229	1 ⊢ (ω ∈ On ∨ ω = On)

Syntax hints:   ∨ wo 844   = wceq 1533   ∈ wcel 2098  Ord word 6354  Oncon0 6355  ωcom 7849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-lim 6360  df-om 7850

This theorem is referenced by:  omelon2  7862  infensuc  9152  elhf2  35669