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Theorem 2lnat 34547
 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 1089 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ HL)
2 hlatl 34124 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
31, 2syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ AtLat)
4 hllat 34127 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
51, 4syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ Lat)
6 simp12 1090 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝑋𝐵)
7 simp13 1091 . . . . 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 16973 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
115, 6, 7, 10syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐵)
12 simp3r 1088 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ≠ 0 )
13 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
14 2lnat.z . . . . 5 0 = (0.‘𝐾)
15 2lnat.a . . . . 5 𝐴 = (Atoms‘𝐾)
168, 13, 14, 15atlex 34080 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑌) ≠ 0 ) → ∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌))
173, 11, 12, 16syl3anc 1323 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → ∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌))
18 simp13l 1174 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑋𝑌)
19 simp11 1089 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵))
20 simp12l 1172 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐹𝑋) ∈ 𝑁)
21 simp12r 1173 . . . . . . . . . 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 34546 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁)) → (𝑋(le‘𝐾)𝑌𝑋 = 𝑌))
2519, 20, 21, 24syl12anc 1321 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋(le‘𝐾)𝑌𝑋 = 𝑌))
26 simp111 1188 . . . . . . . . . . 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 1189 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑋𝐵)
29 simp113 1190 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑌𝐵)
308, 13, 9latleeqm1 17000 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 𝑌) = 𝑋))
3127, 28, 29, 30syl3anc 1323 . . . . . . . . 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 2834 . . . . . . 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 1061 . . . . . . . . 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 16997 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
3727, 28, 29, 36syl3anc 1323 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌)(le‘𝐾)𝑋)
38 hlpos 34129 . . . . . . . . . . 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 34053 . . . . . . . . . . 11 (𝑝𝐴𝑝𝐵)
41403ad2ant2 1081 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝𝐵)
4227, 28, 29, 10syl3anc 1323 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌) ∈ 𝐵)
43 simp2 1060 . . . . . . . . . . 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 16979 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)𝑋)
45 eqid 2621 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
468, 13, 45, 15, 22, 23lncvrat 34545 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) ∧ ((𝐹𝑋) ∈ 𝑁𝑝(le‘𝐾)𝑋)) → 𝑝( ⋖ ‘𝐾)𝑋)
4726, 28, 43, 20, 44, 46syl32anc 1331 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
488, 13, 45cvrnbtwn4 34043 . . . . . . . . . 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 1336 . . . . . . . . 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 955 . . . . . . . 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 2881 . . . . . . . 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 2812 . . . . . 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 1261 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑝𝐴 → (𝑝(le‘𝐾)(𝑋 𝑌) → 𝑝 = (𝑋 𝑌))))
5655reximdvai 3009 . . 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 3055 . 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   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∃wrex 2908   class class class wbr 4613  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  Posetcpo 16861  meetcmee 16866  0.cp0 16958  Latclat 16966   ⋖ ccvr 34026  Atomscatm 34027  AtLatcal 34028  HLchlt 34114  Linesclines 34257  pmapcpmap 34260 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-lines 34264  df-pmap 34267 This theorem is referenced by:  cdleme3h  34999  cdleme7ga  35012
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