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Mirrors > Home > MPE Home > Th. List > ismred2 | Structured version Visualization version GIF version |
Description: Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
Ref | Expression |
---|---|
ismred2.ss | ⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋) |
ismred2.in | ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶) |
Ref | Expression |
---|---|
ismred2 | ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismred2.ss | . 2 ⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋) | |
2 | eqid 2610 | . . . 4 ⊢ ∅ = ∅ | |
3 | rint0 4452 | . . . 4 ⊢ (∅ = ∅ → (𝑋 ∩ ∩ ∅) = 𝑋) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝑋 ∩ ∩ ∅) = 𝑋 |
5 | 0ss 3924 | . . . 4 ⊢ ∅ ⊆ 𝐶 | |
6 | 0ex 4718 | . . . . 5 ⊢ ∅ ∈ V | |
7 | sseq1 3589 | . . . . . . 7 ⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐶 ↔ ∅ ⊆ 𝐶)) | |
8 | 7 | anbi2d 736 | . . . . . 6 ⊢ (𝑠 = ∅ → ((𝜑 ∧ 𝑠 ⊆ 𝐶) ↔ (𝜑 ∧ ∅ ⊆ 𝐶))) |
9 | inteq 4413 | . . . . . . . 8 ⊢ (𝑠 = ∅ → ∩ 𝑠 = ∩ ∅) | |
10 | 9 | ineq2d 3776 | . . . . . . 7 ⊢ (𝑠 = ∅ → (𝑋 ∩ ∩ 𝑠) = (𝑋 ∩ ∩ ∅)) |
11 | 10 | eleq1d 2672 | . . . . . 6 ⊢ (𝑠 = ∅ → ((𝑋 ∩ ∩ 𝑠) ∈ 𝐶 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐶)) |
12 | 8, 11 | imbi12d 333 | . . . . 5 ⊢ (𝑠 = ∅ → (((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶) ↔ ((𝜑 ∧ ∅ ⊆ 𝐶) → (𝑋 ∩ ∩ ∅) ∈ 𝐶))) |
13 | ismred2.in | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶) | |
14 | 6, 12, 13 | vtocl 3232 | . . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐶) → (𝑋 ∩ ∩ ∅) ∈ 𝐶) |
15 | 5, 14 | mpan2 703 | . . 3 ⊢ (𝜑 → (𝑋 ∩ ∩ ∅) ∈ 𝐶) |
16 | 4, 15 | syl5eqelr 2693 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐶) |
17 | simp2 1055 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ⊆ 𝐶) | |
18 | 1 | 3ad2ant1 1075 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋) |
19 | 17, 18 | sstrd 3578 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ⊆ 𝒫 𝑋) |
20 | simp3 1056 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ≠ ∅) | |
21 | rintn0 4552 | . . . 4 ⊢ ((𝑠 ⊆ 𝒫 𝑋 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) = ∩ 𝑠) | |
22 | 19, 20, 21 | syl2anc 691 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) = ∩ 𝑠) |
23 | 13 | 3adant3 1074 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶) |
24 | 22, 23 | eqeltrrd 2689 | . 2 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) |
25 | 1, 16, 24 | ismred 16085 | 1 ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ⊆ 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: isacs1i 16141 mreacs 16142 |
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