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Theorem tron 6387
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 5271 . 2 (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2 vex 3478 . . . . . . 7 𝑥 ∈ V
32elon 6373 . . . . . 6 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordelord 6386 . . . . . 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 6373 . . . 4 (𝑦 ∈ On ↔ Ord 𝑦)
96, 8imbitrrdi 251 . . 3 (𝑥 ∈ On → (𝑦𝑥𝑦 ∈ On))
109ssrdv 3988 . 2 (𝑥 ∈ On → 𝑥 ⊆ On)
111, 10mprgbir 3068 1 Tr On
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wss 3948  Tr wtr 5265  Ord word 6363  Oncon0 6364
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 5299  ax-nul 5306  ax-pr 5427
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  ordon  7766  predon  7775  onuninsuci  7831  gruina  10815
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