Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mreintcl | Structured version Visualization version GIF version |
Description: A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
mreintcl | ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpw2g 4754 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝒫 𝐶 ↔ 𝑆 ⊆ 𝐶)) | |
2 | 1 | biimpar 501 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → 𝑆 ∈ 𝒫 𝐶) |
3 | 2 | 3adant3 1074 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ∈ 𝒫 𝐶) |
4 | ismre 16073 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
5 | 4 | simp3bi 1071 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
6 | 5 | 3ad2ant1 1075 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
7 | simp3 1056 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅) | |
8 | neeq1 2844 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ≠ ∅ ↔ 𝑆 ≠ ∅)) | |
9 | inteq 4413 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ∩ 𝑠 = ∩ 𝑆) | |
10 | 9 | eleq1d 2672 | . . . . 5 ⊢ (𝑠 = 𝑆 → (∩ 𝑠 ∈ 𝐶 ↔ ∩ 𝑆 ∈ 𝐶)) |
11 | 8, 10 | imbi12d 333 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶) ↔ (𝑆 ≠ ∅ → ∩ 𝑆 ∈ 𝐶))) |
12 | 11 | rspcva 3280 | . . 3 ⊢ ((𝑆 ∈ 𝒫 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) → (𝑆 ≠ ∅ → ∩ 𝑆 ∈ 𝐶)) |
13 | 12 | 3impia 1253 | . 2 ⊢ ((𝑆 ∈ 𝒫 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶) |
14 | 3, 6, 7, 13 | syl3anc 1318 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∩ cint 4410 ‘cfv 5804 Moorecmre 16065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-mre 16069 |
This theorem is referenced by: mreiincl 16079 mrerintcl 16080 mreincl 16082 mremre 16087 submre 16088 mrcflem 16089 mrelatglb 17007 mreclatBAD 17010 |
Copyright terms: Public domain | W3C validator |