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| 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 3961 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∖ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
| 2 | df-br 5144 | . 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∖ 𝑆)) | |
| 3 | df-br 5144 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 4 | df-br 5144 | . . . 4 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
| 5 | 4 | notbii 320 | . . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆) |
| 6 | 3, 5 | anbi12i 628 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
| 7 | 1, 2, 6 | 3bitr4i 303 | 1 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∖ cdif 3948 〈cop 4632 class class class wbr 5143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-br 5144 |
| This theorem is referenced by: fundif 6615 fndmdif 7062 isocnv3 7352 brdifun 8775 dflt2 13190 pltval 18377 slenlt 27797 ltgov 28605 opeldifid 32612 qtophaus 33835 dftr6 35751 dffr5 35754 fundmpss 35767 brsset 35890 dfon3 35893 brtxpsd2 35896 dffun10 35915 elfuns 35916 dfrecs2 35951 dfrdg4 35952 dfint3 35953 brub 35955 broutsideof 36122 brvdif 38262 frege124d 43774 |
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