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Mirrors > Home > MPE Home > Th. List > undifabs | Structured version Visualization version GIF version |
Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.) |
Ref | Expression |
---|---|
undifabs | ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undif3 3847 | . 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) | |
2 | unidm 3718 | . . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
3 | 2 | difeq1i 3686 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) = (𝐴 ∖ (𝐵 ∖ 𝐴)) |
4 | difdif 3698 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 | |
5 | 1, 3, 4 | 3eqtri 2636 | 1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∖ cdif 3537 ∪ cun 3538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 |
This theorem is referenced by: dfif5 4052 indifundif 28740 |
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