Theorem ordon 4487

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 4384	. 2 |- Tr On
2		df-on 4370	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2288	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4380	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 121	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2530	. 2 |- A. x e. On Tr x
7		dford3 4369	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 942	1 |- Ord On

Syntax hints:   e. wcel 2148   A.wral 2455   Tr wtr 4103   Ord word 4364   Oncon0 4365

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370

This theorem is referenced by:  ssorduni  4488  limon  4514  onprc  4553  tfri1dALT  6354  rdgon  6389