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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opelvvdif | Structured version Visualization version GIF version |
Description: Negated elementhood of ordered pair. (Contributed by Peter Mazsa, 14-Jan-2019.) |
Ref | Expression |
---|---|
opelvvdif | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ ((V × V) ∖ 𝑅) ↔ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3972 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ((V × V) ∖ 𝑅) ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅)) | |
2 | opelvvg 5729 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
3 | 2 | biantrurd 532 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 ∈ 𝑅 ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅))) |
4 | 1, 3 | bitr4id 290 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ ((V × V) ∖ 𝑅) ↔ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2105 Vcvv 3477 ∖ cdif 3959 〈cop 4636 × cxp 5686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5210 df-xp 5694 |
This theorem is referenced by: vvdifopab 38241 brvvdif 38244 |
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