![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4245 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | ssequn2 4199 | . 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 3961 ∩ cin 3962 ⊆ wss 3963 |
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 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-in 3970 df-ss 3980 |
This theorem is referenced by: volun 25594 |
Copyright terms: Public domain | W3C validator |