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Mirrors > Home > ILE Home > Th. List > difun2 | 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 3329 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) | |
2 | difid 3431 | . . 3 ⊢ (𝐵 ∖ 𝐵) = ∅ | |
3 | 2 | uneq2i 3227 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∪ ∅) |
4 | un0 3396 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ ∅) = (𝐴 ∖ 𝐵) | |
5 | 1, 3, 4 | 3eqtri 2164 | 1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∖ cdif 3068 ∪ cun 3069 ∅c0 3363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 |
This theorem is referenced by: uneqdifeqim 3448 difprsn1 3659 orddif 4462 fisseneq 6820 dfn2 8990 |
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