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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 2799 | . . . 4 ⊢ ∅ = ∅ | |
3 | rint0 4707 | . . . 4 ⊢ (∅ = ∅ → (𝑋 ∩ ∩ ∅) = 𝑋) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝑋 ∩ ∩ ∅) = 𝑋 |
5 | 0ss 4168 | . . . 4 ⊢ ∅ ⊆ 𝐶 | |
6 | 0ex 4984 | . . . . 5 ⊢ ∅ ∈ V | |
7 | sseq1 3822 | . . . . . . 7 ⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐶 ↔ ∅ ⊆ 𝐶)) | |
8 | 7 | anbi2d 623 | . . . . . 6 ⊢ (𝑠 = ∅ → ((𝜑 ∧ 𝑠 ⊆ 𝐶) ↔ (𝜑 ∧ ∅ ⊆ 𝐶))) |
9 | inteq 4670 | . . . . . . . 8 ⊢ (𝑠 = ∅ → ∩ 𝑠 = ∩ ∅) | |
10 | 9 | ineq2d 4012 | . . . . . . 7 ⊢ (𝑠 = ∅ → (𝑋 ∩ ∩ 𝑠) = (𝑋 ∩ ∩ ∅)) |
11 | 10 | eleq1d 2863 | . . . . . 6 ⊢ (𝑠 = ∅ → ((𝑋 ∩ ∩ 𝑠) ∈ 𝐶 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐶)) |
12 | 8, 11 | imbi12d 336 | . . . . 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 3446 | . . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐶) → (𝑋 ∩ ∩ ∅) ∈ 𝐶) |
15 | 5, 14 | mpan2 683 | . . 3 ⊢ (𝜑 → (𝑋 ∩ ∩ ∅) ∈ 𝐶) |
16 | 4, 15 | syl5eqelr 2883 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐶) |
17 | simp2 1168 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ⊆ 𝐶) | |
18 | 1 | 3ad2ant1 1164 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋) |
19 | 17, 18 | sstrd 3808 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ⊆ 𝒫 𝑋) |
20 | simp3 1169 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ≠ ∅) | |
21 | rintn0 4810 | . . . 4 ⊢ ((𝑠 ⊆ 𝒫 𝑋 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) = ∩ 𝑠) | |
22 | 19, 20, 21 | syl2anc 580 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) = ∩ 𝑠) |
23 | 13 | 3adant3 1163 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶) |
24 | 22, 23 | eqeltrrd 2879 | . 2 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) |
25 | 1, 16, 24 | ismred 16577 | 1 ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∩ cin 3768 ⊆ wss 3769 ∅c0 4115 𝒫 cpw 4349 ∩ cint 4667 ‘cfv 6101 Moorecmre 16557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-int 4668 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-iota 6064 df-fun 6103 df-fv 6109 df-mre 16561 |
This theorem is referenced by: isacs1i 16632 mreacs 16633 |
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