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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 2927 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
2 | eldif 2927 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | orbi12i 681 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
4 | elun 3084 | . . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴))) | |
5 | excxor 1269 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) | |
6 | ancom 253 | . . . . 5 ⊢ ((¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
7 | 6 | orbi2i 679 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
8 | 5, 7 | bitri 173 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
9 | 3, 4, 8 | 3bitr4i 201 | . 2 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)) |
10 | 9 | abbi2i 2152 | 1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ∨ wo 629 = wceq 1243 ⊻ wxo 1266 ∈ wcel 1393 {cab 2026 ∖ cdif 2914 ∪ cun 2915 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-xor 1267 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 |
This theorem is referenced by: (None) |
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