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Theorem tron 6384
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 5270 . 2 (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2 vex 3478 . . . . . . 7 𝑥 ∈ V
32elon 6370 . . . . . 6 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordelord 6383 . . . . . 6 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
53, 4sylanb 581 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
65ex 413 . . . 4 (𝑥 ∈ On → (𝑦𝑥 → Ord 𝑦))
7 vex 3478 . . . . 5 𝑦 ∈ V
87elon 6370 . . . 4 (𝑦 ∈ On ↔ Ord 𝑦)
96, 8syl6ibr 251 . . 3 (𝑥 ∈ On → (𝑦𝑥𝑦 ∈ On))
109ssrdv 3987 . 2 (𝑥 ∈ On → 𝑥 ⊆ On)
111, 10mprgbir 3068 1 Tr On
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wss 3947  Tr wtr 5264  Ord word 6360  Oncon0 6361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365
This theorem is referenced by:  ordon  7760  predon  7769  onuninsuci  7825  gruina  10809
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