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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 3217 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) | |
2 | difin 3201 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
3 | incom 3158 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
4 | 3 | difeq2i 3087 | . . . 4 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ (𝐵 ∩ 𝐴)) |
5 | difin 3201 | . . . 4 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴) | |
6 | 4, 5 | eqtri 2101 | . . 3 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴) |
7 | 2, 6 | uneq12i 3124 | . 2 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) |
8 | 1, 7 | eqtr2i 2102 | 1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∖ cdif 2970 ∪ cun 2971 ∩ cin 2972 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 |
This theorem is referenced by: (None) |
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