Theorem omon 7724

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 7722	. 2 ⊢ Ord ω
2		ordeleqon 7632	. 2 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3	1, 2	mpbi 229	1 ⊢ (ω ∈ On ∨ ω = On)

Syntax hints:   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Ord word 6265  Oncon0 6266  ωcom 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-lim 6271  df-om 7713

This theorem is referenced by:  omelon2  7725  infensuc  8942  elhf2  34477