Theorem tron 4417

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 4135	. 2 |- ( Tr On <-> A. x e. On x C_ On )
2		vex 2766	. . . . . . 7 |- x e. _V
3	2	elon 4409	. . . . . 6 |- ( x e. On <-> Ord x )
4		ordelord 4416	. . . . . 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 2766	. . . . 5 |- y e. _V
8	7	elon 4409	. . . 4 |- ( y e. On <-> Ord y )
9	6, 8	imbitrrdi 162	. . 3 |- ( x e. On -> ( y e. x -> y e. On ) )
10	9	ssrdv 3189	. 2 |- ( x e. On -> x C_ On )
11	1, 10	mprgbir 2555	1 |- Tr On

Syntax hints:   e. wcel 2167   C_ wss 3157   Tr wtr 4131   Ord word 4397   Oncon0 4398

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403

This theorem is referenced by:  ordon  4522  tfi  4618