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Mirrors > Home > MPE Home > Th. List > Mathboxes > brvvdif | Structured version Visualization version GIF version |
Description: Binary relation with the complement under the universal class of ordered pairs. (Contributed by Peter Mazsa, 9-Nov-2018.) |
Ref | Expression |
---|---|
brvvdif | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelvvdif 35535 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ ((V × V) ∖ 𝑅) ↔ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅)) | |
2 | df-br 5067 | . 2 ⊢ (𝐴((V × V) ∖ 𝑅)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ((V × V) ∖ 𝑅)) | |
3 | df-br 5067 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | 3 | notbii 322 | . 2 ⊢ (¬ 𝐴𝑅𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅) |
5 | 1, 2, 4 | 3bitr4g 316 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 〈cop 4573 class class class wbr 5066 × cxp 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 |
This theorem is referenced by: brvbrvvdif 35540 |
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