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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 4546 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝐶 ↔ 𝑠 ⊆ 𝐶) | |
4 | ismred.in | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) | |
5 | 4 | 3expia 1117 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
6 | 3, 5 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐶) → (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
7 | 6 | ralrimiva 3184 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)) |
8 | ismre 16863 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
9 | 1, 2, 7, 8 | syl3anbrc 1339 | 1 ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ⊆ 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-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-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: ismred2 16876 mremre 16877 submre 16878 subrgmre 19561 lssmre 19740 cssmre 20839 cldmre 21688 toponmre 21703 ismrcd1 39302 |
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