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Mirrors > Home > ILE Home > Th. List > symdifxor | GIF version |
Description: Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Ref | Expression |
---|---|
symdifxor | ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3046 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
2 | eldif 3046 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | orbi12i 736 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
4 | elun 3183 | . . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴))) | |
5 | excxor 1339 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) | |
6 | ancom 264 | . . . . 5 ⊢ ((¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
7 | 6 | orbi2i 734 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
8 | 5, 7 | bitri 183 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
9 | 3, 4, 8 | 3bitr4i 211 | . 2 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)) |
10 | 9 | abbi2i 2229 | 1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∨ wo 680 = wceq 1314 ⊻ wxo 1336 ∈ wcel 1463 {cab 2101 ∖ cdif 3034 ∪ cun 3035 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-xor 1337 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-dif 3039 df-un 3041 |
This theorem is referenced by: (None) |
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