Theorem tron 4367

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 4091	. 2 |- ( Tr On <-> A. x e. On x C_ On )
2		vex 2733	. . . . . . 7 |- x e. _V
3	2	elon 4359	. . . . . 6 |- ( x e. On <-> Ord x )
4		ordelord 4366	. . . . . 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 2733	. . . . 5 |- y e. _V
8	7	elon 4359	. . . 4 |- ( y e. On <-> Ord y )
9	6, 8	syl6ibr 161	. . 3 |- ( x e. On -> ( y e. x -> y e. On ) )
10	9	ssrdv 3153	. 2 |- ( x e. On -> x C_ On )
11	1, 10	mprgbir 2528	1 |- Tr On

Syntax hints:   e. wcel 2141   C_ wss 3121   Tr wtr 4087   Ord word 4347   Oncon0 4348

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  ordon  4470  tfi  4566