Theorem tron 4304

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 4030	. 2 |- ( Tr On <-> A. x e. On x C_ On )
2		vex 2689	. . . . . . 7 |- x e. _V
3	2	elon 4296	. . . . . 6 |- ( x e. On <-> Ord x )
4		ordelord 4303	. . . . . 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 2689	. . . . 5 |- y e. _V
8	7	elon 4296	. . . 4 |- ( y e. On <-> Ord y )
9	6, 8	syl6ibr 161	. . 3 |- ( x e. On -> ( y e. x -> y e. On ) )
10	9	ssrdv 3103	. 2 |- ( x e. On -> x C_ On )
11	1, 10	mprgbir 2490	1 |- Tr On

Syntax hints:   e. wcel 1480   C_ wss 3071   Tr wtr 4026   Ord word 4284   Oncon0 4285

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290

This theorem is referenced by:  ordon  4402  tfi  4496