MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordon Structured version   Visualization version   GIF version

Theorem ordon 7764
Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
Assertion
Ref Expression
ordon Ord On

Proof of Theorem ordon
StepHypRef Expression
1 tron 6388 . 2 Tr On
2 epweon 7762 . 2 E We On
3 df-ord 6368 . 2 (Ord On ↔ (Tr On ∧ E We On))
41, 2, 3mpbir2an 710 1 Ord On
Colors of variables: wff setvar class
Syntax hints:  Tr wtr 5266   E cep 5580   We wwe 5631  Ord word 6364  Oncon0 6365
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369
This theorem is referenced by:  onprc  7765  ssorduni  7766  ordeleqon  7769  ordsson  7770  onint  7778  ordsuci  7796  sucexeloniOLD  7798  suceloniOLD  7800  limon  7824  tfi  7842  ordom  7865  ordtypelem2  9514  hartogs  9539  card2on  9549  tskwe  9945  alephsmo  10097  ondomon  10558  dford3lem2  41766  dford3  41767  tfsconcatlem  42086  iunord  47721
  Copyright terms: Public domain W3C validator