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Theorem cvrcmp 37683
Description: If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
Hypotheses
Ref Expression
cvrcmp.b 𝐵 = (Base‘𝐾)
cvrcmp.l = (le‘𝐾)
cvrcmp.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrcmp ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))

Proof of Theorem cvrcmp
StepHypRef Expression
1 simpl1 1191 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝐾 ∈ Poset)
2 simpl23 1253 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐵)
3 simpl21 1251 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋𝐵)
4 simpl3l 1228 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑋)
5 cvrcmp.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cvrcmp.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
75, 6cvrne 37681 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍𝑋)
81, 2, 3, 4, 7syl31anc 1373 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝑋)
9 cvrcmp.l . . . . . . . 8 = (le‘𝐾)
105, 9, 6cvrle 37678 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 𝑋)
111, 2, 3, 4, 10syl31anc 1373 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍 𝑋)
12 simpr 485 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 𝑌)
13 simpl22 1252 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑌𝐵)
14 simpl3r 1229 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑌)
155, 9, 6cvrnbtwn4 37679 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑍𝐵𝑌𝐵𝑋𝐵) ∧ 𝑍𝐶𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
161, 2, 13, 3, 14, 15syl131anc 1383 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
1711, 12, 16mpbi2and 710 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍 = 𝑋𝑋 = 𝑌))
18 neor 3034 . . . . 5 ((𝑍 = 𝑋𝑋 = 𝑌) ↔ (𝑍𝑋𝑋 = 𝑌))
1917, 18sylib 217 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍𝑋𝑋 = 𝑌))
208, 19mpd 15 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 = 𝑌)
2120ex 413 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
22 simp1 1136 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23 simp21 1206 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋𝐵)
245, 9posref 18167 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
2522, 23, 24syl2anc 584 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋 𝑋)
26 breq2 5107 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
2725, 26syl5ibcom 244 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 = 𝑌𝑋 𝑌))
2821, 27impbid 211 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2941   class class class wbr 5103  cfv 6493  Basecbs 17043  lecple 17100  Posetcpo 18156  ccvr 37662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-proset 18144  df-poset 18162  df-plt 18179  df-covers 37666
This theorem is referenced by:  cvrcmp2  37684  atcmp  37711  llncmp  37923  lplncmp  37963  lvolcmp  38018  lhp2lt  38402
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