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Mirrors > Home > MPE Home > Th. List > difun2 | Structured version Visualization version GIF version |
Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.) |
Ref | Expression |
---|---|
difun2 | ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difundir 4259 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) | |
2 | difid 4332 | . . 3 ⊢ (𝐵 ∖ 𝐵) = ∅ | |
3 | 2 | uneq2i 4138 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∪ ∅) |
4 | un0 4346 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ ∅) = (𝐴 ∖ 𝐵) | |
5 | 1, 3, 4 | 3eqtri 2850 | 1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3935 ∪ cun 3936 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 |
This theorem is referenced by: uneqdifeq 4440 difprsn1 4735 orddif 6286 domunsncan 8619 elfiun 8896 hartogslem1 9008 cantnfp1lem3 9145 dju1dif 9600 infdju1 9617 ssxr 10712 dfn2 11913 incexclem 15193 mreexmrid 16916 lbsextlem4 19935 ufprim 22519 volun 24148 i1f1 24293 itgioo 24418 itgsplitioo 24440 plyeq0lem 24802 jensen 25568 difeq 30282 fzdif2 30516 fzodif2 30517 pmtrcnel2 30736 measun 31472 carsgclctunlem1 31577 carsggect 31578 chtvalz 31902 elmrsubrn 32769 mrsubvrs 32771 pibt2 34700 finixpnum 34879 lindsadd 34887 lindsenlbs 34889 poimirlem2 34896 poimirlem4 34898 poimirlem6 34900 poimirlem7 34901 poimirlem8 34902 poimirlem11 34905 poimirlem12 34906 poimirlem13 34907 poimirlem14 34908 poimirlem16 34910 poimirlem18 34912 poimirlem19 34913 poimirlem21 34915 poimirlem23 34917 poimirlem27 34921 poimirlem30 34924 asindmre 34979 kelac2 39672 pwfi2f1o 39703 iccdifioo 41798 iccdifprioo 41799 hoiprodp1 42877 |
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