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Theorem tron 4209
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 3940 . 2 (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2 vex 2622 . . . . . . 7 𝑥 ∈ V
32elon 4201 . . . . . 6 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordelord 4208 . . . . . 6 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
53, 4sylanb 278 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
65ex 113 . . . 4 (𝑥 ∈ On → (𝑦𝑥 → Ord 𝑦))
7 vex 2622 . . . . 5 𝑦 ∈ V
87elon 4201 . . . 4 (𝑦 ∈ On ↔ Ord 𝑦)
96, 8syl6ibr 160 . . 3 (𝑥 ∈ On → (𝑦𝑥𝑦 ∈ On))
109ssrdv 3031 . 2 (𝑥 ∈ On → 𝑥 ⊆ On)
111, 10mprgbir 2433 1 Tr On
Colors of variables: wff set class
Syntax hints:  wcel 1438  wss 2999  Tr wtr 3936  Ord word 4189  Oncon0 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by:  ordon  4303  tfi  4397
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