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Mirrors > Home > MPE Home > Th. List > elriin | Structured version Visualization version GIF version |
Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
elriin | ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4169 | . 2 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆)) | |
2 | eliin 4924 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆 ↔ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)) | |
3 | 2 | pm5.32i 577 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)) |
4 | 1, 3 | bitri 277 | 1 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 ∩ ciin 4920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-in 3943 df-iin 4922 |
This theorem is referenced by: limciun 24492 limcun 24493 |
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