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| Mirrors > Home > MPE Home > Th. List > ismred | Structured version Visualization version GIF version | ||
| Description: Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
| Ref | Expression |
|---|---|
| ismred.ss | ⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋) |
| ismred.ba | ⊢ (𝜑 → 𝑋 ∈ 𝐶) |
| ismred.in | ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) |
| Ref | Expression |
|---|---|
| ismred | ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ismred.ss | . 2 ⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋) | |
| 2 | ismred.ba | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐶) | |
| 3 | velpw 4605 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝐶 ↔ 𝑠 ⊆ 𝐶) | |
| 4 | ismred.in | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) | |
| 5 | 4 | 3expia 1122 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
| 6 | 3, 5 | sylan2b 594 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐶) → (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
| 7 | 6 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
| 8 | ismre 17633 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
| 9 | 1, 2, 7, 8 | syl3anbrc 1344 | 1 ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∩ cint 4946 ‘cfv 6561 Moorecmre 17625 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-mre 17629 |
| This theorem is referenced by: ismred2 17646 mremre 17647 submre 17648 subrngmre 20562 subrgmre 20597 lssmre 20964 cssmre 21711 cldmre 23086 toponmre 23101 zartopn 33874 ismrcd1 42709 |
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