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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 4215 | . 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) | |
2 | unidm 4079 | . . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
3 | 2 | difeq1i 4046 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) = (𝐴 ∖ (𝐵 ∖ 𝐴)) |
4 | difdif 4058 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 | |
5 | 1, 3, 4 | 3eqtri 2825 | 1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∖ cdif 3878 ∪ cun 3879 |
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-rab 3115 df-v 3443 df-dif 3884 df-un 3886 |
This theorem is referenced by: dfif5 4441 indifundif 30297 |
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