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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 3867 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ((V × V) ∖ 𝑅) ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅)) | |
2 | opelvvg 5580 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
3 | 2 | biantrurd 536 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 ∈ 𝑅 ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅))) |
4 | 1, 3 | bitr4id 293 | 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 209 ∧ wa 399 ∈ wcel 2110 Vcvv 3401 ∖ cdif 3854 〈cop 4537 × cxp 5538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-opab 5106 df-xp 5546 |
This theorem is referenced by: vvdifopab 36093 brvvdif 36096 |
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