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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 3954 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ((V × V) ∖ 𝑅) ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅)) | |
2 | opelvvg 5709 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
3 | 2 | biantrurd 533 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 ∈ 𝑅 ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑅))) |
4 | 1, 3 | bitr4id 289 | 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 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3473 ∖ cdif 3941 〈cop 4628 × cxp 5667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-opab 5204 df-xp 5675 |
This theorem is referenced by: vvdifopab 36933 brvvdif 36936 |
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