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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 4123 | . 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On) | |
2 | vex 2755 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 2 | elon 4395 | . . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
4 | ordelord 4402 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) | |
5 | 3, 4 | sylanb 284 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) |
6 | 5 | ex 115 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦)) |
7 | vex 2755 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 7 | elon 4395 | . . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦) |
9 | 6, 8 | imbitrrdi 162 | . . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On)) |
10 | 9 | ssrdv 3176 | . 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On) |
11 | 1, 10 | mprgbir 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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