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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 2733 | . . . 4 ⊢ ∅ = ∅ | |
3 | rint0 4993 | . . . 4 ⊢ (∅ = ∅ → (𝑋 ∩ ∩ ∅) = 𝑋) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝑋 ∩ ∩ ∅) = 𝑋 |
5 | 0ss 4395 | . . . 4 ⊢ ∅ ⊆ 𝐶 | |
6 | 0ex 5306 | . . . . 5 ⊢ ∅ ∈ V | |
7 | sseq1 4006 | . . . . . . 7 ⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐶 ↔ ∅ ⊆ 𝐶)) | |
8 | 7 | anbi2d 630 | . . . . . 6 ⊢ (𝑠 = ∅ → ((𝜑 ∧ 𝑠 ⊆ 𝐶) ↔ (𝜑 ∧ ∅ ⊆ 𝐶))) |
9 | inteq 4952 | . . . . . . . 8 ⊢ (𝑠 = ∅ → ∩ 𝑠 = ∩ ∅) | |
10 | 9 | ineq2d 4211 | . . . . . . 7 ⊢ (𝑠 = ∅ → (𝑋 ∩ ∩ 𝑠) = (𝑋 ∩ ∩ ∅)) |
11 | 10 | eleq1d 2819 | . . . . . 6 ⊢ (𝑠 = ∅ → ((𝑋 ∩ ∩ 𝑠) ∈ 𝐶 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐶)) |
12 | 8, 11 | imbi12d 345 | . . . . 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 3549 | . . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐶) → (𝑋 ∩ ∩ ∅) ∈ 𝐶) |
15 | 5, 14 | mpan2 690 | . . 3 ⊢ (𝜑 → (𝑋 ∩ ∩ ∅) ∈ 𝐶) |
16 | 4, 15 | eqeltrrid 2839 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐶) |
17 | simp2 1138 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ⊆ 𝐶) | |
18 | 1 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋) |
19 | 17, 18 | sstrd 3991 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ⊆ 𝒫 𝑋) |
20 | simp3 1139 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → 𝑠 ≠ ∅) | |
21 | rintn0 5111 | . . . 4 ⊢ ((𝑠 ⊆ 𝒫 𝑋 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) = ∩ 𝑠) | |
22 | 19, 20, 21 | syl2anc 585 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) = ∩ 𝑠) |
23 | 13 | 3adant3 1133 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶) |
24 | 22, 23 | eqeltrrd 2835 | . 2 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) |
25 | 1, 16, 24 | ismred 17542 | 1 ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ∩ cint 4949 ‘cfv 6540 Moorecmre 17522 |
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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
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 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-mre 17526 |
This theorem is referenced by: isacs1i 17597 mreacs 17598 |
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