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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 4543 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝐶 ↔ 𝑠 ⊆ 𝐶) | |
4 | ismred.in | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) | |
5 | 4 | 3expia 1113 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
6 | 3, 5 | sylan2b 593 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐶) → (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
7 | 6 | ralrimiva 3179 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
8 | ismre 16849 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
9 | 1, 2, 7, 8 | syl3anbrc 1335 | 1 ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 ∩ cint 4867 ‘cfv 6348 Moorecmre 16841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-mre 16845 |
This theorem is referenced by: ismred2 16862 mremre 16863 submre 16864 subrgmre 19488 lssmre 19667 cssmre 20765 cldmre 21614 toponmre 21629 ismrcd1 39173 |
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