Theorem ordon 4534

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 4429	. 2 |- Tr On
2		df-on 4415	. . . . 5 |- On = { x | Ord x }
3	2	abeq2i 2316	. . . 4 |- ( x e. On <-> Ord x )
4		ordtr 4425	. . . 4 |- ( Ord x -> Tr x )
5	3, 4	sylbi 121	. . 3 |- ( x e. On -> Tr x )
6	5	rgen 2559	. 2 |- A. x e. On Tr x
7		dford3 4414	. 2 |- ( Ord On <-> ( Tr On /\ A. x e. On Tr x ) )
8	1, 6, 7	mpbir2an 945	1 |- Ord On

Syntax hints:   e. wcel 2176   A.wral 2484   Tr wtr 4142   Ord word 4409   Oncon0 4410

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by:  ssorduni  4535  limon  4561  onprc  4600  tfri1dALT  6437  rdgon  6472