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Theorem cvrcmp 36730
 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 1188 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝐾 ∈ Poset)
2 simpl23 1250 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐵)
3 simpl21 1248 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋𝐵)
4 simpl3l 1225 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑋)
5 cvrcmp.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cvrcmp.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
75, 6cvrne 36728 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍𝑋)
81, 2, 3, 4, 7syl31anc 1370 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝑋)
9 cvrcmp.l . . . . . . . 8 = (le‘𝐾)
105, 9, 6cvrle 36725 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 𝑋)
111, 2, 3, 4, 10syl31anc 1370 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍 𝑋)
12 simpr 488 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 𝑌)
13 simpl22 1249 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑌𝐵)
14 simpl3r 1226 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑌)
155, 9, 6cvrnbtwn4 36726 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑍𝐵𝑌𝐵𝑋𝐵) ∧ 𝑍𝐶𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
161, 2, 13, 3, 14, 15syl131anc 1380 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
1711, 12, 16mpbi2and 711 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍 = 𝑋𝑋 = 𝑌))
18 neor 3078 . . . . 5 ((𝑍 = 𝑋𝑋 = 𝑌) ↔ (𝑍𝑋𝑋 = 𝑌))
1917, 18sylib 221 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍𝑋𝑋 = 𝑌))
208, 19mpd 15 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 = 𝑌)
2120ex 416 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
22 simp1 1133 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23 simp21 1203 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋𝐵)
245, 9posref 17573 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
2522, 23, 24syl2anc 587 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋 𝑋)
26 breq2 5038 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
2725, 26syl5ibcom 248 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 = 𝑌𝑋 𝑌))
2821, 27impbid 215 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   class class class wbr 5034  ‘cfv 6332  Basecbs 16495  lecple 16584  Posetcpo 17562   ⋖ ccvr 36709 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6291  df-fun 6334  df-fv 6340  df-proset 17550  df-poset 17568  df-plt 17580  df-covers 36713 This theorem is referenced by:  cvrcmp2  36731  atcmp  36758  llncmp  36969  lplncmp  37009  lvolcmp  37064  lhp2lt  37448
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