Theorem tron 5746

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 4756	. 2 |- ( Tr On <-> A. x e. On x C_ On )
2		vex 3203	. . . . . . 7 |- x e. _V
3	2	elon 5732	. . . . . 6 |- ( x e. On <-> Ord x )
4		ordelord 5745	. . . . . 6 |- ( ( Ord x /\ y e. x ) -> Ord y )
5	3, 4	sylanb 489	. . . . 5 |- ( ( x e. On /\ y e. x ) -> Ord y )
6	5	ex 450	. . . 4 |- ( x e. On -> ( y e. x -> Ord y ) )
7		vex 3203	. . . . 5 |- y e. _V
8	7	elon 5732	. . . 4 |- ( y e. On <-> Ord y )
9	6, 8	syl6ibr 242	. . 3 |- ( x e. On -> ( y e. x -> y e. On ) )
10	9	ssrdv 3609	. 2 |- ( x e. On -> x C_ On )
11	1, 10	mprgbir 2927	1 |- Tr On

Syntax hints:   e. wcel 1990   C_ wss 3574   Tr wtr 4752   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  ordon  6982  onuninsuci  7040  gruina  9640