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Definition df-cv 28310
Description: Define the covers relation (on the Hilbert lattice). Definition 3.2.18 of [PtakPulmannova] p. 68, whose notation we use. Ptak/Pulmannova's notation 𝐴 𝐵 is read "𝐵 covers 𝐴 " or "𝐴 is covered by 𝐵 " , and it means that 𝐵 is larger than 𝐴 and there is nothing in between. See cvbr 28313 and cvbr2 28314 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
df-cv = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ (𝑥𝑦 ∧ ¬ ∃𝑧C (𝑥𝑧𝑧𝑦)))}
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-cv
StepHypRef Expression
1 ccv 26993 . 2 class
2 vx . . . . . . 7 setvar 𝑥
32cv 1473 . . . . . 6 class 𝑥
4 cch 26958 . . . . . 6 class C
53, 4wcel 1938 . . . . 5 wff 𝑥C
6 vy . . . . . . 7 setvar 𝑦
76cv 1473 . . . . . 6 class 𝑦
87, 4wcel 1938 . . . . 5 wff 𝑦C
95, 8wa 382 . . . 4 wff (𝑥C𝑦C )
103, 7wpss 3445 . . . . 5 wff 𝑥𝑦
11 vz . . . . . . . . . 10 setvar 𝑧
1211cv 1473 . . . . . . . . 9 class 𝑧
133, 12wpss 3445 . . . . . . . 8 wff 𝑥𝑧
1412, 7wpss 3445 . . . . . . . 8 wff 𝑧𝑦
1513, 14wa 382 . . . . . . 7 wff (𝑥𝑧𝑧𝑦)
1615, 11, 4wrex 2801 . . . . . 6 wff 𝑧C (𝑥𝑧𝑧𝑦)
1716wn 3 . . . . 5 wff ¬ ∃𝑧C (𝑥𝑧𝑧𝑦)
1810, 17wa 382 . . . 4 wff (𝑥𝑦 ∧ ¬ ∃𝑧C (𝑥𝑧𝑧𝑦))
199, 18wa 382 . . 3 wff ((𝑥C𝑦C ) ∧ (𝑥𝑦 ∧ ¬ ∃𝑧C (𝑥𝑧𝑧𝑦)))
2019, 2, 6copab 4540 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ (𝑥𝑦 ∧ ¬ ∃𝑧C (𝑥𝑧𝑧𝑦)))}
211, 20wceq 1474 1 wff = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ (𝑥𝑦 ∧ ¬ ∃𝑧C (𝑥𝑧𝑧𝑦)))}
Colors of variables: wff setvar class
This definition is referenced by:  cvbr  28313
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