Theorem tron 4137

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 3879	. 2 |- ( Tr On <-> A. x e. On x C_ On )
2		vex 2604	. . . . . . 7 |- x e. _V
3	2	elon 4129	. . . . . 6 |- ( x e. On <-> Ord x )
4		ordelord 4136	. . . . . 6 |- ( ( Ord x /\ y e. x ) -> Ord y )
5	3, 4	sylanb 278	. . . . 5 |- ( ( x e. On /\ y e. x ) -> Ord y )
6	5	ex 113	. . . 4 |- ( x e. On -> ( y e. x -> Ord y ) )
7		vex 2604	. . . . 5 |- y e. _V
8	7	elon 4129	. . . 4 |- ( y e. On <-> Ord y )
9	6, 8	syl6ibr 160	. . 3 |- ( x e. On -> ( y e. x -> y e. On ) )
10	9	ssrdv 3005	. 2 |- ( x e. On -> x C_ On )
11	1, 10	mprgbir 2421	1 |- Tr On

Syntax hints:   e. wcel 1433   C_ wss 2973   Tr wtr 3875   Ord word 4117   Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by:  ordon  4230  tfi  4323