![]() |
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 5344 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝒫 𝐶 ↔ 𝑆 ⊆ 𝐶)) | |
2 | 1 | biimpar 479 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → 𝑆 ∈ 𝒫 𝐶) |
3 | 2 | 3adant3 1133 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ∈ 𝒫 𝐶) |
4 | ismre 17531 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
5 | 4 | simp3bi 1148 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
6 | 5 | 3ad2ant1 1134 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
7 | simp3 1139 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅) | |
8 | neeq1 3004 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ≠ ∅ ↔ 𝑆 ≠ ∅)) | |
9 | inteq 4953 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ∩ 𝑠 = ∩ 𝑆) | |
10 | 9 | eleq1d 2819 | . . . . 5 ⊢ (𝑠 = 𝑆 → (∩ 𝑠 ∈ 𝐶 ↔ ∩ 𝑆 ∈ 𝐶)) |
11 | 8, 10 | imbi12d 345 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶) ↔ (𝑆 ≠ ∅ → ∩ 𝑆 ∈ 𝐶))) |
12 | 11 | rspcva 3611 | . . 3 ⊢ ((𝑆 ∈ 𝒫 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) → (𝑆 ≠ ∅ → ∩ 𝑆 ∈ 𝐶)) |
13 | 12 | 3impia 1118 | . 2 ⊢ ((𝑆 ∈ 𝒫 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶) |
14 | 3, 6, 7, 13 | syl3anc 1372 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∩ cint 4950 ‘cfv 6541 Moorecmre 17523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6493 df-fun 6543 df-fv 6549 df-mre 17527 |
This theorem is referenced by: mreiincl 17537 mrerintcl 17538 mreincl 17540 mremre 17545 submre 17546 mrcflem 17547 mrelatglb 18510 mreclatBAD 18513 mrelatglbALT 47575 mreclat 47576 |
Copyright terms: Public domain | W3C validator |