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Theorem cdleme0nex 39674
Description: Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p ∨ q/0 (i.e. the sublattice from 0 to p ∨ q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 39595- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 38726, our (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ) is a shorter way to express π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄). Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)
Hypotheses
Ref Expression
cdleme0nex.l ≀ = (leβ€˜πΎ)
cdleme0nex.j ∨ = (joinβ€˜πΎ)
cdleme0nex.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cdleme0nex (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
Distinct variable groups:   𝐴,π‘Ÿ   ∨ ,π‘Ÿ   ≀ ,π‘Ÿ   𝑃,π‘Ÿ   𝑄,π‘Ÿ   𝑅,π‘Ÿ   π‘Š,π‘Ÿ
Allowed substitution hint:   𝐾(π‘Ÿ)

Proof of Theorem cdleme0nex
StepHypRef Expression
1 simp3r 1199 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ Β¬ 𝑅 ≀ π‘Š)
2 simp12 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄))
31, 2jca 511 . . 3 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)))
4 simp3l 1198 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ 𝑅 ∈ 𝐴)
5 simp13 1202 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))
6 ralnex 3066 . . . . . . 7 (βˆ€π‘Ÿ ∈ 𝐴 Β¬ (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ↔ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))
75, 6sylibr 233 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ βˆ€π‘Ÿ ∈ 𝐴 Β¬ (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))
8 breq1 5144 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ ≀ π‘Š ↔ 𝑅 ≀ π‘Š))
98notbid 318 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (Β¬ π‘Ÿ ≀ π‘Š ↔ Β¬ 𝑅 ≀ π‘Š))
10 oveq2 7413 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (𝑃 ∨ π‘Ÿ) = (𝑃 ∨ 𝑅))
11 oveq2 7413 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (𝑄 ∨ π‘Ÿ) = (𝑄 ∨ 𝑅))
1210, 11eqeq12d 2742 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ ((𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
139, 12anbi12d 630 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ((Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ↔ (Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))))
1413notbid 318 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Β¬ (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ↔ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))))
1514rspcva 3604 . . . . . 6 ((𝑅 ∈ 𝐴 ∧ βˆ€π‘Ÿ ∈ 𝐴 Β¬ (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) β†’ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
164, 7, 15syl2anc 583 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
17 simp11 1200 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ 𝐾 ∈ HL)
18 hlcvl 38742 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ CvLat)
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ 𝐾 ∈ CvLat)
20 simp21 1203 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ 𝑃 ∈ 𝐴)
21 simp22 1204 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ 𝑄 ∈ 𝐴)
22 simp23 1205 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ 𝑃 β‰  𝑄)
23 cdleme0nex.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
24 cdleme0nex.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
25 cdleme0nex.j . . . . . . . 8 ∨ = (joinβ€˜πΎ)
2623, 24, 25cvlsupr2 38726 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
2719, 20, 21, 4, 22, 26syl131anc 1380 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
2827anbi2d 628 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ ((Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ↔ (Β¬ 𝑅 ≀ π‘Š ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)))))
2916, 28mtbid 324 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
30 ianor 978 . . . . 5 (Β¬ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∧ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) ↔ (Β¬ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∨ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
31 df-3an 1086 . . . . . . . 8 ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)) ↔ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)))
3231anbi2i 622 . . . . . . 7 ((Β¬ 𝑅 ≀ π‘Š ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) ↔ (Β¬ 𝑅 ≀ π‘Š ∧ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
33 an12 642 . . . . . . 7 ((Β¬ 𝑅 ≀ π‘Š ∧ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) ↔ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∧ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
3432, 33bitri 275 . . . . . 6 ((Β¬ 𝑅 ≀ π‘Š ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) ↔ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∧ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
3534notbii 320 . . . . 5 (Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) ↔ Β¬ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∧ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
36 pm4.62 853 . . . . 5 (((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) β†’ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) ↔ (Β¬ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∨ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
3730, 35, 363bitr4ri 304 . . . 4 (((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) β†’ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) ↔ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
3829, 37sylibr 233 . . 3 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) β†’ Β¬ (Β¬ 𝑅 ≀ π‘Š ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
393, 38mt2d 136 . 2 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ Β¬ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄))
40 neanior 3029 . . 3 ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ↔ Β¬ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
4140con2bii 357 . 2 ((𝑅 = 𝑃 ∨ 𝑅 = 𝑄) ↔ Β¬ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄))
4239, 41sylibr 233 1 (((𝐾 ∈ HL ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  lecple 17213  joincjn 18276  Atomscatm 38646  CvLatclc 38648  HLchlt 38733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734
This theorem is referenced by:  cdleme18c  39677  cdleme18d  39679  cdlemg17b  40046  cdlemg17h  40052
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