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Theorem tron 4085
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.
StepHypRef Expression
1 dftr3 3849 . 2 (Tr On ↔ x On x ⊆ On)
2 vex 2554 . . . . . . 7 x V
32elon 4077 . . . . . 6 (x On ↔ Ord x)
4 ordelord 4084 . . . . . 6 ((Ord x y x) → Ord y)
53, 4sylanb 268 . . . . 5 ((x On y x) → Ord y)
65ex 108 . . . 4 (x On → (y x → Ord y))
7 vex 2554 . . . . 5 y V
87elon 4077 . . . 4 (y On ↔ Ord y)
96, 8syl6ibr 151 . . 3 (x On → (y xy On))
109ssrdv 2945 . 2 (x On → x ⊆ On)
111, 10mprgbir 2373 1 Tr On
Colors of variables: wff set class
Syntax hints:   wcel 1390  wss 2911  Tr wtr 3845  Ord word 4065  Oncon0 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by:  ordon  4178  tfi  4248
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