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Theorem tron 4403
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 4123 . 2 (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2 vex 2755 . . . . . . 7 𝑥 ∈ V
32elon 4395 . . . . . 6 (𝑥 ∈ On ↔ Ord 𝑥)
4 ordelord 4402 . . . . . 6 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
53, 4sylanb 284 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
65ex 115 . . . 4 (𝑥 ∈ On → (𝑦𝑥 → Ord 𝑦))
7 vex 2755 . . . . 5 𝑦 ∈ V
87elon 4395 . . . 4 (𝑦 ∈ On ↔ Ord 𝑦)
96, 8imbitrrdi 162 . . 3 (𝑥 ∈ On → (𝑦𝑥𝑦 ∈ On))
109ssrdv 3176 . 2 (𝑥 ∈ On → 𝑥 ⊆ On)
111, 10mprgbir 2548 1 Tr On
Colors of variables: wff set class
Syntax hints:  wcel 2160  wss 3144  Tr wtr 4119  Ord word 4383  Oncon0 4384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3828  df-tr 4120  df-iord 4387  df-on 4389
This theorem is referenced by:  ordon  4506  tfi  4602
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