Theorem tron 4360

Description: The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)

Assertion

Ref	Expression
tron	|- Tr On

Proof of Theorem tron

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr3 4084	. 2 |- ( Tr On <-> A. x e. On x C_ On )
2		vex 2729	. . . . . . 7 |- x e. _V
3	2	elon 4352	. . . . . 6 |- ( x e. On <-> Ord x )
4		ordelord 4359	. . . . . 6 |- ( ( Ord x /\ y e. x ) -> Ord y )
5	3, 4	sylanb 282	. . . . 5 |- ( ( x e. On /\ y e. x ) -> Ord y )
6	5	ex 114	. . . 4 |- ( x e. On -> ( y e. x -> Ord y ) )
7		vex 2729	. . . . 5 |- y e. _V
8	7	elon 4352	. . . 4 |- ( y e. On <-> Ord y )
9	6, 8	syl6ibr 161	. . 3 |- ( x e. On -> ( y e. x -> y e. On ) )
10	9	ssrdv 3148	. 2 |- ( x e. On -> x C_ On )
11	1, 10	mprgbir 2524	1 |- Tr On

Syntax hints:   e. wcel 2136   C_ wss 3116   Tr wtr 4080   Ord word 4340   Oncon0 4341

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  ordon  4463  tfi  4559