Theorem ordon 4633

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 4527	. 2 |- Tr On
2		df-on 4513	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2349	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4523	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 121	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2603	. 2 |- A. x e. On Tr x
7		dford3 4512	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 955	1 |- Ord On

This proof depends on syntax axioms:   e. wcel 2209   A.wral 2528   Tr wtr 4229   Ord word 4507   Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513

This theorem is used by:  ssorduni  4634  limon  4660  onprc  4699  tfri1dALT  6622  rdgon  6657