![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > brdif | Structured version Visualization version 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 3954 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∖ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 5150 | . 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∖ 𝑆)) | |
3 | df-br 5150 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 5150 | . . . 4 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 4 | notbii 319 | . . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆) |
6 | 3, 5 | anbi12i 626 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
7 | 1, 2, 6 | 3bitr4i 302 | 1 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∖ cdif 3941 〈cop 4636 class class class wbr 5149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-dif 3947 df-br 5150 |
This theorem is referenced by: fundif 6603 fndmdif 7050 isocnv3 7339 brdifun 8754 dflt2 13162 pltval 18327 slenlt 27731 ltgov 28473 opeldifid 32468 qtophaus 33568 dftr6 35476 dffr5 35479 fundmpss 35493 brsset 35616 dfon3 35619 brtxpsd2 35622 dffun10 35641 elfuns 35642 dfrecs2 35677 dfrdg4 35678 dfint3 35679 brub 35681 broutsideof 35848 brvdif 37863 frege124d 43333 |
Copyright terms: Public domain | W3C validator |