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Theorem omon 4847
Description: The class of natural numbers  om 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  |-  ( om  e.  On  \/  om  =  On )

Proof of Theorem omon
StepHypRef Expression
1 ordom 4845 . 2  |-  Ord  om
2 ordeleqon 4760 . 2  |-  ( Ord 
om 
<->  ( om  e.  On  \/  om  =  On ) )
31, 2mpbi 200 1  |-  ( om  e.  On  \/  om  =  On )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1652    e. wcel 1725   Ord word 4572   Oncon0 4573   omcom 4836
This theorem is referenced by:  omelon2  4848  infensuc  7276  elhf2  26064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837
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