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Theorem cdleme0nex 38782
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 38703- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 37834, 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 1203 . . . 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 1205 . . . 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 513 . . 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 1202 . . . . . 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 1206 . . . . . . 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 3076 . . . . . . 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 5113 . . . . . . . . . 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 7370 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (𝑃 ∨ π‘Ÿ) = (𝑃 ∨ 𝑅))
11 oveq2 7370 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (𝑄 ∨ π‘Ÿ) = (𝑄 ∨ 𝑅))
1210, 11eqeq12d 2753 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ ((𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
139, 12anbi12d 632 . . . . . . . 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 3582 . . . . . 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 585 . . . . 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 1204 . . . . . . . 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 37850 . . . . . . . 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 1207 . . . . . . 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 1208 . . . . . . 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 1209 . . . . . . 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 37834 . . . . . . 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 1384 . . . . . 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 630 . . . . 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 981 . . . . 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 1090 . . . . . . . 8 ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)) ↔ ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)))
3231anbi2i 624 . . . . . . 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 644 . . . . . . 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 pm4.62 855 . . . . 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 3038 . . 3 ((𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄) ↔ Β¬ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
4140con2bii 358 . 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  lecple 17147  joincjn 18207  Atomscatm 37754  CvLatclc 37756  HLchlt 37841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-proset 18191  df-poset 18209  df-plt 18226  df-lub 18242  df-glb 18243  df-join 18244  df-meet 18245  df-p0 18321  df-lat 18328  df-covers 37757  df-ats 37758  df-atl 37789  df-cvlat 37813  df-hlat 37842
This theorem is referenced by:  cdleme18c  38785  cdleme18d  38787  cdlemg17b  39154  cdlemg17h  39160
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