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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 5265 | . 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On) | |
| 2 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 3 | 2 | elon 6393 | . . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
| 4 | ordelord 6406 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) | |
| 5 | 3, 4 | sylanb 581 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) |
| 6 | 5 | ex 412 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦)) |
| 7 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 8 | 7 | elon 6393 | . . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦) |
| 9 | 6, 8 | imbitrrdi 252 | . . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On)) |
| 10 | 9 | ssrdv 3989 | . 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On) |
| 11 | 1, 10 | mprgbir 3068 | 1 ⊢ Tr On |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ⊆ wss 3951 Tr wtr 5259 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: ordon 7797 predon 7806 onuninsuci 7861 gruina 10858 |
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