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Theorem tron 4299
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 4025 . 2 (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2 vex 2684 . . . . . . 7 𝑥 ∈ V
32elon 4291 . . . . . 6 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordelord 4298 . . . . . 6 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
53, 4sylanb 282 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
65ex 114 . . . 4 (𝑥 ∈ On → (𝑦𝑥 → Ord 𝑦))
7 vex 2684 . . . . 5 𝑦 ∈ V
87elon 4291 . . . 4 (𝑦 ∈ On ↔ Ord 𝑦)
96, 8syl6ibr 161 . . 3 (𝑥 ∈ On → (𝑦𝑥𝑦 ∈ On))
109ssrdv 3098 . 2 (𝑥 ∈ On → 𝑥 ⊆ On)
111, 10mprgbir 2488 1 Tr On
Colors of variables: wff set class
Syntax hints:  wcel 1480  wss 3066  Tr wtr 4021  Ord word 4279  Oncon0 4280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285
This theorem is referenced by:  ordon  4397  tfi  4491
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