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Theorem ordon 7482
 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 6186 . 2 Tr On
2 epweon 7481 . 2 E We On
3 df-ord 6166 . 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 5139   E cep 5432   We wwe 5481  Ord word 6162  Oncon0 6163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167 This theorem is referenced by:  onprc  7483  ssorduni  7484  ordeleqon  7487  ordsson  7488  onint  7494  suceloni  7512  limon  7535  tfi  7552  ordom  7573  ordtypelem2  8971  hartogs  8996  card2on  9006  tskwe  9367  alephsmo  9517  ondomon  9978  dford3lem2  39961  dford3  39962  iunord  45199
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