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Mirrors > Home > MPE Home > Th. List > Mathboxes > brvdif | Structured version Visualization version GIF version |
Description: Binary relation with universal complement is the negation of the relation. (Contributed by Peter Mazsa, 1-Jul-2018.) |
Ref | Expression |
---|---|
brvdif | ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brv 5468 | . 2 ⊢ 𝐴V𝐵 | |
2 | brdif 5195 | . 2 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ (𝐴V𝐵 ∧ ¬ 𝐴𝑅𝐵)) | |
3 | 1, 2 | mpbiran 708 | 1 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 Vcvv 3469 ∖ cdif 3941 class class class wbr 5142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 |
This theorem is referenced by: brvdif2 37669 brvbrvvdif 37671 dfssr2 37908 |
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