| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iinssiin | Structured version Visualization version GIF version | ||
| Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| iinssiin.1 | ⊢ Ⅎ𝑥𝜑 |
| iinssiin.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
| Ref | Expression |
|---|---|
| iinssiin | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iinssiin.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
| 2 | nfii1 5027 | . . . . . 6 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 | |
| 3 | 2 | nfcri 2896 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 |
| 4 | 1, 3 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) |
| 5 | iinssiin.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) | |
| 6 | 5 | adantlr 715 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
| 7 | eliinid 45089 | . . . . . . 7 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) | |
| 8 | 7 | adantll 714 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) |
| 9 | 6, 8 | sseldd 3983 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐶) |
| 10 | 9 | ex 412 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐶)) |
| 11 | 4, 10 | ralrimi 3256 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) |
| 12 | eliin 4994 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
| 13 | 12 | elv 3484 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) |
| 14 | 11, 13 | sylibr 234 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) |
| 15 | 14 | ssd 45058 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3060 Vcvv 3479 ⊆ wss 3950 ∩ ciin 4990 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-v 3481 df-ss 3967 df-iin 4992 |
| This theorem is referenced by: (None) |
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