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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 4292 | . 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) | |
2 | unidm 4152 | . . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
3 | 2 | difeq1i 4117 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) = (𝐴 ∖ (𝐵 ∖ 𝐴)) |
4 | difdif 4130 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 | |
5 | 1, 3, 4 | 3eqtri 2758 | 1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∖ cdif 3944 ∪ cun 3945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 |
This theorem is referenced by: dfif5 4549 indifundif 32451 |
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