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Mirrors > Home > MPE Home > Th. List > tron | Structured version Visualization version 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 5179 | . 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On) | |
2 | vex 3424 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 2 | elon 6239 | . . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
4 | ordelord 6252 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) | |
5 | 3, 4 | sylanb 584 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) |
6 | 5 | ex 416 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦)) |
7 | vex 3424 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 7 | elon 6239 | . . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦) |
9 | 6, 8 | syl6ibr 255 | . . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On)) |
10 | 9 | ssrdv 3921 | . 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On) |
11 | 1, 10 | mprgbir 3077 | 1 ⊢ Tr On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ⊆ wss 3880 Tr wtr 5175 Ord word 6229 Oncon0 6230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-11 2159 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-ral 3067 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-tr 5176 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-ord 6233 df-on 6234 |
This theorem is referenced by: ordon 7579 predon 7587 onuninsuci 7637 gruina 10456 |
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