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Theorem ordon 7716
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 6345 . 2 Tr On
2 epweon 7714 . 2 E We On
3 df-ord 6325 . 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 5227   E cep 5541   We wwe 5592  Ord word 6321  Oncon0 6322
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  onprc  7717  ssorduni  7718  ordeleqon  7721  ordsson  7722  onint  7730  ordsuci  7748  sucexeloniOLD  7750  suceloniOLD  7752  limon  7776  tfi  7794  ordom  7817  ordtypelem2  9462  hartogs  9487  card2on  9497  tskwe  9893  alephsmo  10045  ondomon  10506  dford3lem2  41380  dford3  41381  iunord  47195
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