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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 4293 | . 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) | |
2 | unidm 4153 | . . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
3 | 2 | difeq1i 4118 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) = (𝐴 ∖ (𝐵 ∖ 𝐴)) |
4 | difdif 4131 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 | |
5 | 1, 3, 4 | 3eqtri 2760 | 1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3946 ∪ cun 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 |
This theorem is referenced by: dfif5 4548 indifundif 32342 |
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