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Mirrors > Home > MPE Home > Th. List > rintn0 | Structured version Visualization version GIF version |
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 4904 | . . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴) | |
2 | ssid 3992 | . . . 4 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴 | |
3 | sspwuni 5025 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴) | |
4 | 2, 3 | mpbi 232 | . . 3 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴 |
5 | 1, 4 | sstrdi 3982 | . 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ 𝐴) |
6 | sseqin2 4195 | . 2 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) | |
7 | 5, 6 | sylib 220 | 1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ≠ wne 3019 ∩ cin 3938 ⊆ wss 3939 ∅c0 4294 𝒫 cpw 4542 ∪ cuni 4841 ∩ cint 4879 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 df-nul 4295 df-pw 4544 df-uni 4842 df-int 4880 |
This theorem is referenced by: mrerintcl 16871 ismred2 16877 |
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