Theorem ordon 4582

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 4477	. 2 |- Tr On
2		df-on 4463	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2340	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4473	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 121	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2583	. 2 |- A. x e. On Tr x
7		dford3 4462	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 948	1 |- Ord On

Syntax hints:   e. wcel 2200   A.wral 2508   Tr wtr 4185   Ord word 4457   Oncon0 4458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-in 3204  df-ss 3211  df-uni 3892  df-tr 4186  df-iord 4461  df-on 4463

This theorem is referenced by:  ssorduni  4583  limon  4609  onprc  4648  tfri1dALT  6512  rdgon  6547