Theorem ordon 7731

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

Step	Hyp	Ref	Expression
1		tron 6346	. 2 ⊢ Tr On
2		epweon 7729	. 2 ⊢ E We On
3		df-ord 6326	. 2 ⊢ (Ord On ↔ (Tr On ∧ E We On))
4	1, 2, 3	mpbir2an 712	1 ⊢ Ord On

Syntax hints:  Tr wtr 5192   E cep 5530   We wwe 5583  Ord word 6322  Oncon0 6323

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327

This theorem is referenced by:  onprc  7732  ssorduni  7733  ordeleqon  7736  ordsson  7737  onint  7744  ordsuci  7762  limon  7787  tfi  7804  ordom  7827  ordtypelem2  9434  hartogs  9459  card2on  9469  tskwe  9874  alephsmo  10024  ondomon  10485  dford3lem2  43455  dford3  43456  tfsconcatlem  43764  iunord  50151