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| Mirrors > Home > MPE Home > Th. List > unabs | Structured version Visualization version GIF version | ||
| Description: Absorption law for union. (Contributed by NM, 16-Apr-2006.) |
| Ref | Expression |
|---|---|
| unabs | ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | inss1 4237 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
| 2 | ssequn2 4189 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴) | |
| 3 | 1, 2 | mpbi 230 | 1 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 |
| This theorem is referenced by: volun 25580 |
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