Theorem ordon 4230

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 4137	. 2 |- Tr On
2		df-on 4123	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2189	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4133	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 119	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2416	. 2 |- A. x e. On Tr x
7		dford3 4122	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 883	1 |- Ord On

Syntax hints:   e. wcel 1433   A.wral 2348   Tr wtr 3875   Ord word 4117   Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by:  ssorduni  4231  limon  4257  onprc  4295