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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 5169 | . 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On) | |
2 | vex 3498 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 2 | elon 6195 | . . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
4 | ordelord 6208 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) | |
5 | 3, 4 | sylanb 583 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦) |
6 | 5 | ex 415 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦)) |
7 | vex 3498 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | 7 | elon 6195 | . . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦) |
9 | 6, 8 | syl6ibr 254 | . . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On)) |
10 | 9 | ssrdv 3973 | . 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On) |
11 | 1, 10 | mprgbir 3153 | 1 ⊢ Tr On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ⊆ wss 3936 Tr wtr 5165 Ord word 6185 Oncon0 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-ord 6189 df-on 6190 |
This theorem is referenced by: ordon 7492 onuninsuci 7549 gruina 10234 |
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