Theorem ordon 4552

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 4447	. 2 |- Tr On
2		df-on 4433	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2318	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4443	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 121	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2561	. 2 |- A. x e. On Tr x
7		dford3 4432	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 945	1 |- Ord On

Syntax hints:   e. wcel 2178   A.wral 2486   Tr wtr 4158   Ord word 4427   Oncon0 4428

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433

This theorem is referenced by:  ssorduni  4553  limon  4579  onprc  4618  tfri1dALT  6460  rdgon  6495