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Theorem tron 6253
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 5179 . 2 (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2 vex 3424 . . . . . . 7 𝑥 ∈ V
32elon 6239 . . . . . 6 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordelord 6252 . . . . . 6 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
53, 4sylanb 584 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
65ex 416 . . . 4 (𝑥 ∈ On → (𝑦𝑥 → Ord 𝑦))
7 vex 3424 . . . . 5 𝑦 ∈ V
87elon 6239 . . . 4 (𝑦 ∈ On ↔ Ord 𝑦)
96, 8syl6ibr 255 . . 3 (𝑥 ∈ On → (𝑦𝑥𝑦 ∈ On))
109ssrdv 3921 . 2 (𝑥 ∈ On → 𝑥 ⊆ On)
111, 10mprgbir 3077 1 Tr On
Colors of variables: wff setvar class
Syntax hints:  wcel 2111  wss 3880  Tr wtr 5175  Ord word 6229  Oncon0 6230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-11 2159  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2942  df-ral 3067  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-tr 5176  df-eprel 5474  df-po 5482  df-so 5483  df-fr 5523  df-we 5525  df-ord 6233  df-on 6234
This theorem is referenced by:  ordon  7579  predon  7587  onuninsuci  7637  gruina  10456
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