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Mirrors > Home > MPE Home > Th. List > triin | Structured version Visualization version GIF version |
Description: An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.) |
Ref | Expression |
---|---|
triin | ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3386 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | eliin 4713 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
4 | r19.26 3243 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
5 | 4 | biimpri 220 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵)) |
6 | trss 4952 | . . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵)) | |
7 | 6 | imp 396 | . . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵) |
8 | 7 | ralimi 3131 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵) |
10 | ssiin 4758 | . . . . 5 ⊢ (𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵) | |
11 | 9, 10 | sylibr 226 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵) |
12 | 3, 11 | sylan2b 588 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵) |
13 | 12 | ralrimiva 3145 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵) |
14 | dftr3 4947 | . 2 ⊢ (Tr ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵) | |
15 | 13, 14 | sylibr 226 | 1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∈ wcel 2157 ∀wral 3087 Vcvv 3383 ⊆ wss 3767 ∩ ciin 4709 Tr wtr 4943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-v 3385 df-in 3774 df-ss 3781 df-uni 4627 df-iin 4711 df-tr 4944 |
This theorem is referenced by: trint 4959 |
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