Theorem omon 7898

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 7896	. 2 ⊢ Ord ω
2		ordeleqon 7800	. 2 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3	1, 2	mpbi 230	1 ⊢ (ω ∈ On ∨ ω = On)

Syntax hints:   ∨ wo 847   = wceq 1536   ∈ wcel 2105  Ord word 6384  Oncon0 6385  ωcom 7886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-om 7887

This theorem is referenced by:  omelon2  7899  infensuc  9193  elhf2  36156