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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 4903 | . . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴) | |
2 | ssid 3991 | . . . 4 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴 | |
3 | sspwuni 5024 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴) | |
4 | 2, 3 | mpbi 232 | . . 3 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴 |
5 | 1, 4 | sstrdi 3981 | . 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ 𝐴) |
6 | sseqin2 4194 | . 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 1537 ≠ wne 3018 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 ∪ cuni 4840 ∩ cint 4878 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-pw 4543 df-uni 4841 df-int 4879 |
This theorem is referenced by: mrerintcl 16870 ismred2 16876 |
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