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Mirrors > Home > ILE Home > Th. List > symdif1 | GIF version |
Description: Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
symdif1 | ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difundir 3380 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) | |
2 | difin 3364 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
3 | incom 3319 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
4 | 3 | difeq2i 3242 | . . . 4 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ (𝐵 ∩ 𝐴)) |
5 | difin 3364 | . . . 4 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴) | |
6 | 4, 5 | eqtri 2191 | . . 3 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴) |
7 | 2, 6 | uneq12i 3279 | . 2 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) |
8 | 1, 7 | eqtr2i 2192 | 1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∖ cdif 3118 ∪ cun 3119 ∩ cin 3120 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 |
This theorem is referenced by: (None) |
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