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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 2921 | . . . 4 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B)) | |
2 | eldif 2921 | . . . 4 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A)) | |
3 | 1, 2 | orbi12i 680 | . . 3 ⊢ ((x ∈ (A ∖ B) ∨ x ∈ (B ∖ A)) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∨ (x ∈ B ∧ ¬ x ∈ A))) |
4 | elun 3078 | . . 3 ⊢ (x ∈ ((A ∖ B) ∪ (B ∖ A)) ↔ (x ∈ (A ∖ B) ∨ x ∈ (B ∖ A))) | |
5 | excxor 1268 | . . . 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 678 | . . . 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 2149 | 1 ⊢ ((A ∖ B) ∪ (B ∖ A)) = {x ∣ (x ∈ A ⊻ x ∈ B)} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ∨ wo 628 = wceq 1242 ⊻ wxo 1265 ∈ wcel 1390 {cab 2023 ∖ cdif 2908 ∪ cun 2909 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-xor 1266 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-dif 2914 df-un 2916 |
This theorem is referenced by: (None) |
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