![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 4863 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ⊆ ∪ 𝐴) | |
4 | 1, 2, 3 | syl2anc 587 | . . 3 ⊢ (𝜑 → ∩ 𝐵 ⊆ ∪ 𝐴) |
5 | sseqin2 4142 | . . 3 ⊢ (∩ 𝐵 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵) | |
6 | 4, 5 | sylib 221 | . 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵) |
7 | bj-ismoored.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Moore) | |
8 | 7, 1 | bj-ismoored 34522 | . 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴) |
9 | 6, 8 | eqeltrrd 2891 | 1 ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 ∩ cint 4838 Moorecmoore 34518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-uni 4801 df-int 4839 df-bj-moore 34519 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |