Theorem tron 4427

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 4145	. 2 |- ( Tr On <-> A. x e. On x C_ On )
2		vex 2774	. . . . . . 7 |- x e. _V
3	2	elon 4419	. . . . . 6 |- ( x e. On <-> Ord x )
4		ordelord 4426	. . . . . 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 2774	. . . . 5 |- y e. _V
8	7	elon 4419	. . . 4 |- ( y e. On <-> Ord y )
9	6, 8	imbitrrdi 162	. . 3 |- ( x e. On -> ( y e. x -> y e. On ) )
10	9	ssrdv 3198	. 2 |- ( x e. On -> x C_ On )
11	1, 10	mprgbir 2563	1 |- Tr On

Syntax hints:   e. wcel 2175   C_ wss 3165   Tr wtr 4141   Ord word 4407   Oncon0 4408

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-in 3171  df-ss 3178  df-uni 3850  df-tr 4142  df-iord 4411  df-on 4413

This theorem is referenced by:  ordon  4532  tfi  4628