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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 4297 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) | |
2 | difid 4382 | . . 3 ⊢ (𝐵 ∖ 𝐵) = ∅ | |
3 | 2 | uneq2i 4175 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∪ ∅) |
4 | un0 4400 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ ∅) = (𝐴 ∖ 𝐵) | |
5 | 1, 3, 4 | 3eqtri 2767 | 1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3960 ∪ cun 3961 ∅c0 4339 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-nul 4340 |
This theorem is referenced by: undif5 4491 uneqdifeq 4499 difprsn1 4805 orddif 6482 domunsncan 9111 elfiun 9468 hartogslem1 9580 cantnfp1lem3 9718 dju1dif 10211 infdju1 10228 ssxr 11328 dfn2 12537 incexclem 15869 mreexmrid 17688 lbsextlem4 21181 ufprim 23933 volun 25594 i1f1 25739 itgioo 25866 itgsplitioo 25888 plyeq0lem 26264 jensen 27047 difeq 32546 fzdif2 32799 fzodif2 32800 pmtrcnel2 33093 measun 34192 carsgclctunlem1 34299 carsggect 34300 chtvalz 34623 elmrsubrn 35505 mrsubvrs 35507 pibt2 37400 finixpnum 37592 lindsadd 37600 lindsenlbs 37602 poimirlem2 37609 poimirlem4 37611 poimirlem6 37613 poimirlem7 37614 poimirlem8 37615 poimirlem11 37618 poimirlem12 37619 poimirlem13 37620 poimirlem14 37621 poimirlem16 37623 poimirlem18 37625 poimirlem19 37626 poimirlem21 37628 poimirlem23 37630 poimirlem27 37634 poimirlem30 37637 asindmre 37690 disjresundif 38224 kelac2 43054 pwfi2f1o 43085 iccdifioo 45468 iccdifprioo 45469 hoiprodp1 46544 |
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