Theorem omon 7857

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 7855	. 2 ⊢ Ord ω
2		ordeleqon 7761	. 2 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3	1, 2	mpbi 230	1 ⊢ (ω ∈ On ∨ ω = On)

Syntax hints:   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Ord word 6334  Oncon0 6335  ωcom 7845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339  df-lim 6340  df-om 7846

This theorem is referenced by:  omelon2  7858  infensuc  9125  elhf2  36170