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Mathbox for Glauco Siliprandi |
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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 4747 | . . . 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 224 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
6 | eliind.d | . . 3 ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)) | |
7 | 6 | rspcva 3524 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐷) |
8 | 1, 5, 7 | syl2anc 579 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ∩ ciin 4743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-v 3416 df-iin 4745 |
This theorem is referenced by: iooiinioc 40572 hspdifhsp 41618 smflimlem3 41769 smfsuplem1 41805 smflimsuplem4 41817 |
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