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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 4737 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ⊆ ∪ 𝐴) | |
4 | 1, 2, 3 | syl2anc 579 | . . 3 ⊢ (𝜑 → ∩ 𝐵 ⊆ ∪ 𝐴) |
5 | sseqin2 4040 | . . 3 ⊢ (∩ 𝐵 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵) | |
6 | 4, 5 | sylib 210 | . 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵) |
7 | bj-ismoored.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Moore) | |
8 | 7, 1 | bj-ismoored 33643 | . 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴) |
9 | 6, 8 | eqeltrrd 2860 | 1 ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 ∪ cuni 4673 ∩ cint 4712 Moorecmoore 33638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 ax-sep 5019 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-v 3400 df-dif 3795 df-in 3799 df-ss 3806 df-nul 4142 df-pw 4381 df-uni 4674 df-int 4713 df-bj-moore 33639 |
This theorem is referenced by: (None) |
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