Theorem ordon 4410

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 4312	. 2 |- Tr On
2		df-on 4298	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2251	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4308	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 120	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2488	. 2 |- A. x e. On Tr x
7		dford3 4297	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 927	1 |- Ord On

Syntax hints:   e. wcel 1481   A.wral 2417   Tr wtr 4034   Ord word 4292   Oncon0 4293

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298

This theorem is referenced by:  ssorduni  4411  limon  4437  onprc  4475  tfri1dALT  6256  rdgon  6291