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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 2823 | . . . 4 ⊢ ∅ = ∅ | |
3 | rint0 4918 | . . . 4 ⊢ (∅ = ∅ → (𝑋 ∩ ∩ ∅) = 𝑋) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝑋 ∩ ∩ ∅) = 𝑋 |
5 | 0ss 4352 | . . . 4 ⊢ ∅ ⊆ 𝐶 | |
6 | 0ex 5213 | . . . . 5 ⊢ ∅ ∈ V | |
7 | sseq1 3994 | . . . . . . 7 ⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐶 ↔ ∅ ⊆ 𝐶)) | |
8 | 7 | anbi2d 630 | . . . . . 6 ⊢ (𝑠 = ∅ → ((𝜑 ∧ 𝑠 ⊆ 𝐶) ↔ (𝜑 ∧ ∅ ⊆ 𝐶))) |
9 | inteq 4881 | . . . . . . . 8 ⊢ (𝑠 = ∅ → ∩ 𝑠 = ∩ ∅) | |
10 | 9 | ineq2d 4191 | . . . . . . 7 ⊢ (𝑠 = ∅ → (𝑋 ∩ ∩ 𝑠) = (𝑋 ∩ ∩ ∅)) |
11 | 10 | eleq1d 2899 | . . . . . 6 ⊢ (𝑠 = ∅ → ((𝑋 ∩ ∩ 𝑠) ∈ 𝐶 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐶)) |
12 | 8, 11 | imbi12d 347 | . . . . 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 3561 | . . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐶) → (𝑋 ∩ ∩ ∅) ∈ 𝐶) |
15 | 5, 14 | mpan2 689 | . . 3 ⊢ (𝜑 → (𝑋 ∩ ∩ ∅) ∈ 𝐶) |
16 | 4, 15 | eqeltrrid 2920 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐶) |
17 | simp2 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ⊆ 𝐶) | |
18 | 1 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋) |
19 | 17, 18 | sstrd 3979 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ⊆ 𝒫 𝑋) |
20 | simp3 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ≠ ∅) | |
21 | rintn0 5032 | . . . 4 ⊢ ((𝑠 ⊆ 𝒫 𝑋 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) = ∩ 𝑠) | |
22 | 19, 20, 21 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) = ∩ 𝑠) |
23 | 13 | 3adant3 1128 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶) |
24 | 22, 23 | eqeltrrd 2916 | . 2 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) |
25 | 1, 16, 24 | ismred 16875 | 1 ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 ∩ cint 4878 ‘cfv 6357 Moorecmre 16855 |
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 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-mre 16859 |
This theorem is referenced by: isacs1i 16930 mreacs 16931 |
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