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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 | opelvvg 5590 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
2 | 1 | biantrurd 535 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 ∈ 𝑅 ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅))) |
3 | eldif 3946 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ((V × V) ∖ 𝑅) ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅)) | |
4 | 2, 3 | syl6rbbr 292 | 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 208 ∧ wa 398 ∈ wcel 2110 Vcvv 3495 ∖ cdif 3933 〈cop 4567 × cxp 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-opab 5122 df-xp 5556 |
This theorem is referenced by: vvdifopab 35515 brvvdif 35518 |
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