Theorem ordon 4447

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 4344	. 2 |- Tr On
2		df-on 4330	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2268	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4340	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 120	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2510	. 2 |- A. x e. On Tr x
7		dford3 4329	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 927	1 |- Ord On

Syntax hints:   e. wcel 2128   A.wral 2435   Tr wtr 4064   Ord word 4324   Oncon0 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3775  df-tr 4065  df-iord 4328  df-on 4330

This theorem is referenced by:  ssorduni  4448  limon  4474  onprc  4513  tfri1dALT  6300  rdgon  6335