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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ismoored2 | Structured version Visualization version GIF version |
Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
Ref | Expression |
---|---|
bj-ismoored.1 | ⊢ (𝜑 → 𝐴 ∈ Moore) |
bj-ismoored.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
bj-ismoored2.3 | ⊢ (𝜑 → 𝐵 ≠ ∅) |
Ref | Expression |
---|---|
bj-ismoored2 | ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ismoored.2 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
2 | bj-ismoored2.3 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅) | |
3 | intssuni2 4978 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ⊆ ∪ 𝐴) | |
4 | 1, 2, 3 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∩ 𝐵 ⊆ ∪ 𝐴) |
5 | sseqin2 4231 | . . 3 ⊢ (∩ 𝐵 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵) | |
6 | 4, 5 | sylib 218 | . 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵) |
7 | bj-ismoored.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Moore) | |
8 | 7, 1 | bj-ismoored 37090 | . 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴) |
9 | 6, 8 | eqeltrrd 2840 | 1 ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 ∩ cint 4951 Moorecmoore 37086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 df-pw 4607 df-uni 4913 df-int 4952 df-bj-moore 37087 |
This theorem is referenced by: (None) |
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