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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliind | Structured version Visualization version GIF version |
Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
eliind.a | ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶) |
eliind.k | ⊢ (𝜑 → 𝐾 ∈ 𝐵) |
eliind.d | ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)) |
Ref | Expression |
---|---|
eliind | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliind.k | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝐵) | |
2 | eliind.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶) | |
3 | eliin 4461 | . . . 4 ⊢ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
5 | 2, 4 | mpbid 221 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
6 | eliind.d | . . 3 ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)) | |
7 | 6 | rspcva 3280 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐷) |
8 | 1, 5, 7 | syl2anc 691 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∩ ciin 4456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-iin 4458 |
This theorem is referenced by: iooiinioc 38630 hspdifhsp 39506 smflimlem3 39659 |
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