Theorem ordon 4402

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 4304	. 2 |- Tr On
2		df-on 4290	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2250	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4300	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 120	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2485	. 2 |- A. x e. On Tr x
7		dford3 4289	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 926	1 |- Ord On

Syntax hints:   e. wcel 1480   A.wral 2416   Tr wtr 4026   Ord word 4284   Oncon0 4285

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290

This theorem is referenced by:  ssorduni  4403  limon  4429  onprc  4467  tfri1dALT  6248  rdgon  6283