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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 3839 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) | |
2 | difid 3902 | . . 3 ⊢ (𝐵 ∖ 𝐵) = ∅ | |
3 | 2 | uneq2i 3726 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∪ ∅) |
4 | un0 3919 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ ∅) = (𝐴 ∖ 𝐵) | |
5 | 1, 3, 4 | 3eqtri 2636 | 1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∖ cdif 3537 ∪ cun 3538 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: uneqdifeq 4009 uneqdifeqOLD 4010 difprsn1 4271 orddif 5737 domunsncan 7945 elfiun 8219 hartogslem1 8330 cantnfp1lem3 8460 cda1dif 8881 infcda1 8898 ssxr 9986 dfn2 11182 incexclem 14407 mreexmrid 16126 lbsextlem4 18982 ufprim 21523 volun 23120 i1f1 23263 itgioo 23388 itgsplitioo 23410 plyeq0lem 23770 jensen 24515 difeq 28739 fzdif2 28939 measun 29601 carsgclctunlem1 29706 carsggect 29707 elmrsubrn 30671 mrsubvrs 30673 finixpnum 32564 lindsenlbs 32574 poimirlem2 32581 poimirlem4 32583 poimirlem6 32585 poimirlem7 32586 poimirlem8 32587 poimirlem11 32590 poimirlem12 32591 poimirlem13 32592 poimirlem14 32593 poimirlem16 32595 poimirlem18 32597 poimirlem19 32598 poimirlem21 32600 poimirlem23 32602 poimirlem27 32606 poimirlem30 32609 asindmre 32665 kelac2 36653 pwfi2f1o 36684 iccdifioo 38588 iccdifprioo 38589 hoiprodp1 39478 |
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