Theorem tron 4479

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 4191	. 2 |- ( Tr On <-> A. x e. On x C_ On )
2		vex 2805	. . . . . . 7 |- x e. _V
3	2	elon 4471	. . . . . 6 |- ( x e. On <-> Ord x )
4		ordelord 4478	. . . . . 6 |- ( ( Ord x /\ y e. x ) -> Ord y )
5	3, 4	sylanb 284	. . . . 5 |- ( ( x e. On /\ y e. x ) -> Ord y )
6	5	ex 115	. . . 4 |- ( x e. On -> ( y e. x -> Ord y ) )
7		vex 2805	. . . . 5 |- y e. _V
8	7	elon 4471	. . . 4 |- ( y e. On <-> Ord y )
9	6, 8	imbitrrdi 162	. . 3 |- ( x e. On -> ( y e. x -> y e. On ) )
10	9	ssrdv 3233	. 2 |- ( x e. On -> x C_ On )
11	1, 10	mprgbir 2590	1 |- Tr On

Syntax hints:   e. wcel 2202   C_ wss 3200   Tr wtr 4187   Ord word 4459   Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  ordon  4584  tfi  4680