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Theorem 2lnat 35593
Description: Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
Hypotheses
Ref Expression
2lnat.b 𝐵 = (Base‘𝐾)
2lnat.m = (meet‘𝐾)
2lnat.z 0 = (0.‘𝐾)
2lnat.a 𝐴 = (Atoms‘𝐾)
2lnat.n 𝑁 = (Lines‘𝐾)
2lnat.f 𝐹 = (pmap‘𝐾)
Assertion
Ref Expression
2lnat (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)

Proof of Theorem 2lnat
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simp11 1245 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ HL)
2 hlatl 35170 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
31, 2syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ AtLat)
4 hllat 35173 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
51, 4syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ Lat)
6 simp12 1246 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝑋𝐵)
7 simp13 1247 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝑌𝐵)
8 2lnat.b . . . . . 6 𝐵 = (Base‘𝐾)
9 2lnat.m . . . . . 6 = (meet‘𝐾)
108, 9latmcl 17261 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
115, 6, 7, 10syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐵)
12 simp3r 1244 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ≠ 0 )
13 eqid 2771 . . . . 5 (le‘𝐾) = (le‘𝐾)
14 2lnat.z . . . . 5 0 = (0.‘𝐾)
15 2lnat.a . . . . 5 𝐴 = (Atoms‘𝐾)
168, 13, 14, 15atlex 35126 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑌) ≠ 0 ) → ∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌))
173, 11, 12, 16syl3anc 1476 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → ∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌))
18 simp13l 1372 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑋𝑌)
19 simp11 1245 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵))
20 simp12l 1370 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐹𝑋) ∈ 𝑁)
21 simp12r 1371 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐹𝑌) ∈ 𝑁)
22 2lnat.n . . . . . . . . . . 11 𝑁 = (Lines‘𝐾)
23 2lnat.f . . . . . . . . . . 11 𝐹 = (pmap‘𝐾)
248, 13, 22, 23lncmp 35592 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁)) → (𝑋(le‘𝐾)𝑌𝑋 = 𝑌))
2519, 20, 21, 24syl12anc 1474 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋(le‘𝐾)𝑌𝑋 = 𝑌))
26 simp111 1386 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ HL)
2726, 4syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ Lat)
28 simp112 1387 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑋𝐵)
29 simp113 1388 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑌𝐵)
308, 13, 9latleeqm1 17288 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 𝑌) = 𝑋))
3127, 28, 29, 30syl3anc 1476 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 𝑌) = 𝑋))
3225, 31bitr3d 270 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 = 𝑌 ↔ (𝑋 𝑌) = 𝑋))
3332necon3bid 2987 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋𝑌 ↔ (𝑋 𝑌) ≠ 𝑋))
3418, 33mpbid 222 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌) ≠ 𝑋)
35 simp3 1132 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌))
368, 13, 9latmle1 17285 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
3727, 28, 29, 36syl3anc 1476 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌)(le‘𝐾)𝑋)
38 hlpos 35175 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3926, 38syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ Poset)
408, 15atbase 35099 . . . . . . . . . . 11 (𝑝𝐴𝑝𝐵)
41403ad2ant2 1128 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝𝐵)
4227, 28, 29, 10syl3anc 1476 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌) ∈ 𝐵)
43 simp2 1131 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝𝐴)
448, 13, 27, 41, 42, 28, 35, 37lattrd 17267 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)𝑋)
45 eqid 2771 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
468, 13, 45, 15, 22, 23lncvrat 35591 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) ∧ ((𝐹𝑋) ∈ 𝑁𝑝(le‘𝐾)𝑋)) → 𝑝( ⋖ ‘𝐾)𝑋)
4726, 28, 43, 20, 44, 46syl32anc 1484 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
488, 13, 45cvrnbtwn4 35089 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑝𝐵𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) ∧ 𝑝( ⋖ ‘𝐾)𝑋) → ((𝑝(le‘𝐾)(𝑋 𝑌) ∧ (𝑋 𝑌)(le‘𝐾)𝑋) ↔ (𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋)))
4939, 41, 28, 42, 47, 48syl131anc 1489 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → ((𝑝(le‘𝐾)(𝑋 𝑌) ∧ (𝑋 𝑌)(le‘𝐾)𝑋) ↔ (𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋)))
5035, 37, 49mpbi2and 685 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋))
51 neor 3034 . . . . . . . 8 ((𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋) ↔ (𝑝 ≠ (𝑋 𝑌) → (𝑋 𝑌) = 𝑋))
5250, 51sylib 208 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑝 ≠ (𝑋 𝑌) → (𝑋 𝑌) = 𝑋))
5352necon1d 2965 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → ((𝑋 𝑌) ≠ 𝑋𝑝 = (𝑋 𝑌)))
5434, 53mpd 15 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝 = (𝑋 𝑌))
55543exp 1112 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑝𝐴 → (𝑝(le‘𝐾)(𝑋 𝑌) → 𝑝 = (𝑋 𝑌))))
5655reximdvai 3163 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌) → ∃𝑝𝐴 𝑝 = (𝑋 𝑌)))
5717, 56mpd 15 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → ∃𝑝𝐴 𝑝 = (𝑋 𝑌))
58 risset 3210 . 2 ((𝑋 𝑌) ∈ 𝐴 ↔ ∃𝑝𝐴 𝑝 = (𝑋 𝑌))
5957, 58sylibr 224 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 828  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062   class class class wbr 4787  cfv 6032  (class class class)co 6794  Basecbs 16065  lecple 16157  Posetcpo 17149  meetcmee 17154  0.cp0 17246  Latclat 17254  ccvr 35072  Atomscatm 35073  AtLatcal 35074  HLchlt 35160  Linesclines 35303  pmapcpmap 35306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-preset 17137  df-poset 17155  df-plt 17167  df-lub 17183  df-glb 17184  df-join 17185  df-meet 17186  df-p0 17248  df-lat 17255  df-clat 17317  df-oposet 34986  df-ol 34988  df-oml 34989  df-covers 35076  df-ats 35077  df-atl 35108  df-cvlat 35132  df-hlat 35161  df-lines 35310  df-pmap 35313
This theorem is referenced by:  cdleme3h  36045  cdleme7ga  36058
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