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Mirrors > Home > ILE Home > Th. List > tron | GIF version |
Description: The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.) |
Ref | Expression |
---|---|
tron | ⊢ Tr On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr3 4000 | . 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On) | |
2 | vex 2663 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 2 | elon 4266 | . . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
4 | ordelord 4273 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) | |
5 | 3, 4 | sylanb 282 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) |
6 | 5 | ex 114 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦)) |
7 | vex 2663 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 7 | elon 4266 | . . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦) |
9 | 6, 8 | syl6ibr 161 | . . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On)) |
10 | 9 | ssrdv 3073 | . 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On) |
11 | 1, 10 | mprgbir 2467 | 1 ⊢ Tr On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 ⊆ wss 3041 Tr wtr 3996 Ord word 4254 Oncon0 4255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 |
This theorem is referenced by: ordon 4372 tfi 4466 |
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