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Mirrors > Home > ILE Home > Th. List > undifabs | 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 | ssid 3044 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | difss 3126 | . . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | unssi 3175 | . 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 |
4 | ssun1 3163 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐴 ∖ 𝐵)) | |
5 | 3, 4 | eqssi 3041 | 1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 ∖ cdif 2996 ∪ cun 2997 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 |
This theorem is referenced by: (None) |
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