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Theorem ordon 7751
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 6379 . 2 Tr On
2 epweon 7749 . 2 E We On
3 df-ord 6359 . 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 5261   E cep 5575   We wwe 5626  Ord word 6355  Oncon0 6356
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 5295  ax-nul 5302  ax-pr 5423
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 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-tr 5262  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6359  df-on 6360
This theorem is referenced by:  onprc  7752  ssorduni  7753  ordeleqon  7756  ordsson  7757  onint  7765  ordsuci  7783  sucexeloniOLD  7785  suceloniOLD  7787  limon  7811  tfi  7829  ordom  7852  ordtypelem2  9501  hartogs  9526  card2on  9536  tskwe  9932  alephsmo  10084  ondomon  10545  dford3lem2  41637  dford3  41638  tfsconcatlem  41957  iunord  47561
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