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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 5270 | . 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On) | |
2 | vex 3478 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 2 | elon 6370 | . . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
4 | ordelord 6383 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) | |
5 | 3, 4 | sylanb 581 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) |
6 | 5 | ex 413 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦)) |
7 | vex 3478 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 7 | elon 6370 | . . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦) |
9 | 6, 8 | syl6ibr 251 | . . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On)) |
10 | 9 | ssrdv 3987 | . 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On) |
11 | 1, 10 | mprgbir 3068 | 1 ⊢ Tr On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ⊆ wss 3947 Tr wtr 5264 Ord word 6360 Oncon0 6361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 |
This theorem is referenced by: ordon 7760 predon 7769 onuninsuci 7825 gruina 10809 |
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