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Theorem ordon 7798
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 6406 . 2 Tr On
2 epweon 7796 . 2 E We On
3 df-ord 6386 . 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 5258   E cep 5582   We wwe 5635  Ord word 6382  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by:  onprc  7799  ssorduni  7800  ordeleqon  7803  ordsson  7804  onint  7811  ordsuci  7829  sucexeloniOLD  7831  suceloniOLD  7833  limon  7857  tfi  7875  ordom  7898  ordtypelem2  9560  hartogs  9585  card2on  9595  tskwe  9991  alephsmo  10143  ondomon  10604  dford3lem2  43044  dford3  43045  tfsconcatlem  43354  iunord  49250
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