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Theorem tron 4146
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 3885 . 2 (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2 vex 2577 . . . . . . 7 𝑥 ∈ V
32elon 4138 . . . . . 6 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordelord 4145 . . . . . 6 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
53, 4sylanb 272 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
65ex 112 . . . 4 (𝑥 ∈ On → (𝑦𝑥 → Ord 𝑦))
7 vex 2577 . . . . 5 𝑦 ∈ V
87elon 4138 . . . 4 (𝑦 ∈ On ↔ Ord 𝑦)
96, 8syl6ibr 155 . . 3 (𝑥 ∈ On → (𝑦𝑥𝑦 ∈ On))
109ssrdv 2978 . 2 (𝑥 ∈ On → 𝑥 ⊆ On)
111, 10mprgbir 2396 1 Tr On
Colors of variables: wff set class
Syntax hints:  wcel 1409  wss 2944  Tr wtr 3881  Ord word 4126  Oncon0 4127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2951  df-ss 2958  df-uni 3608  df-tr 3882  df-iord 4130  df-on 4132
This theorem is referenced by:  ordon  4239  tfi  4332
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