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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 4265 | . 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) | |
2 | unidm 4128 | . . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
3 | 2 | difeq1i 4095 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) = (𝐴 ∖ (𝐵 ∖ 𝐴)) |
4 | difdif 4107 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 | |
5 | 1, 3, 4 | 3eqtri 2848 | 1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3933 ∪ cun 3934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 |
This theorem is referenced by: dfif5 4483 indifundif 30279 |
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