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Mirrors > Home > ILE Home > Th. List > brdif | GIF version |
Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.) |
Ref | Expression |
---|---|
brdif | ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3085 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∖ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 3938 | . 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∖ 𝑆)) | |
3 | df-br 3938 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 3938 | . . . 4 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 4 | notbii 658 | . . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆) |
6 | 3, 5 | anbi12i 456 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
7 | 1, 2, 6 | 3bitr4i 211 | 1 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∈ wcel 1481 ∖ cdif 3073 〈cop 3535 class class class wbr 3937 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-br 3938 |
This theorem is referenced by: fndmdif 5533 brdifun 6464 |
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