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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 | eliin 4999 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
2 | 1 | elv 3469 | . . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
3 | r19.26 3101 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
4 | trss 5272 | . . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵)) | |
5 | 4 | imp 405 | . . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵) |
6 | 5 | ralimi 3073 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵) |
7 | 3, 6 | sylbir 234 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵) |
8 | ssiin 5056 | . . . . 5 ⊢ (𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵) | |
9 | 7, 8 | sylibr 233 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵) |
10 | 2, 9 | sylan2b 592 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵) |
11 | 10 | ralrimiva 3136 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵) |
12 | dftr3 5267 | . 2 ⊢ (Tr ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵) | |
13 | 11, 12 | sylibr 233 | 1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 Vcvv 3463 ⊆ wss 3947 ∩ ciin 4995 Tr wtr 5261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-v 3465 df-ss 3964 df-uni 4907 df-iin 4997 df-tr 5262 |
This theorem is referenced by: trint 5279 |
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