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Mirrors > Home > MPE Home > Th. List > inabs | Structured version Visualization version GIF version |
Description: Absorption law for intersection. (Contributed by NM, 16-Apr-2006.) |
Ref | Expression |
---|---|
inabs | ⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4099 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | df-ss 3898 | . 2 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴) | |
3 | 1, 2 | mpbi 233 | 1 ⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 |
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-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 |
This theorem is referenced by: dfif5 4441 caragenuncllem 43151 |
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