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Theorem ordon 7796
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 6409 . 2 Tr On
2 epweon 7794 . 2 E We On
3 df-ord 6389 . 2 (Ord On ↔ (Tr On ∧ E We On))
41, 2, 3mpbir2an 711 1 Ord On
Colors of variables: wff setvar class
Syntax hints:  Tr wtr 5265   E cep 5588   We wwe 5640  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onprc  7797  ssorduni  7798  ordeleqon  7801  ordsson  7802  onint  7810  ordsuci  7828  sucexeloniOLD  7830  suceloniOLD  7832  limon  7856  tfi  7874  ordom  7897  ordtypelem2  9557  hartogs  9582  card2on  9592  tskwe  9988  alephsmo  10140  ondomon  10601  dford3lem2  43016  dford3  43017  tfsconcatlem  43326  iunord  48907
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