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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 4258 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | ssequn2 4212 | . 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 1537 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-v 3490 df-un 3981 df-in 3983 df-ss 3993 |
This theorem is referenced by: volun 25599 |
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