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| Description: Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.) | 
| Ref | Expression | 
|---|---|
| rintn0 | ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | intssuni2 4973 | . . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴) | |
| 2 | ssid 4006 | . . . 4 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴 | |
| 3 | sspwuni 5100 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴) | |
| 4 | 2, 3 | mpbi 230 | . . 3 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴 | 
| 5 | 1, 4 | sstrdi 3996 | . 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ 𝐴) | 
| 6 | sseqin2 4223 | . 2 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) | |
| 7 | 5, 6 | sylib 218 | 1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 | 
| 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-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-uni 4908 df-int 4947 | 
| This theorem is referenced by: mrerintcl 17640 ismred2 17646 | 
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