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Theorem cvrcmp 34047
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 1062 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝐾 ∈ Poset)
2 simpl23 1139 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐵)
3 simpl21 1137 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋𝐵)
4 simpl3l 1114 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑋)
5 cvrcmp.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cvrcmp.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
75, 6cvrne 34045 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍𝑋)
81, 2, 3, 4, 7syl31anc 1326 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝑋)
9 cvrcmp.l . . . . . . . 8 = (le‘𝐾)
105, 9, 6cvrle 34042 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 𝑋)
111, 2, 3, 4, 10syl31anc 1326 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍 𝑋)
12 simpr 477 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 𝑌)
13 simpl22 1138 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑌𝐵)
14 simpl3r 1115 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑌)
155, 9, 6cvrnbtwn4 34043 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑍𝐵𝑌𝐵𝑋𝐵) ∧ 𝑍𝐶𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
161, 2, 13, 3, 14, 15syl131anc 1336 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
1711, 12, 16mpbi2and 955 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍 = 𝑋𝑋 = 𝑌))
18 neor 2881 . . . . 5 ((𝑍 = 𝑋𝑋 = 𝑌) ↔ (𝑍𝑋𝑋 = 𝑌))
1917, 18sylib 208 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍𝑋𝑋 = 𝑌))
208, 19mpd 15 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 = 𝑌)
2120ex 450 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
22 simp1 1059 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23 simp21 1092 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋𝐵)
245, 9posref 16872 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
2522, 23, 24syl2anc 692 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋 𝑋)
26 breq2 4617 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
2725, 26syl5ibcom 235 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 = 𝑌𝑋 𝑌))
2821, 27impbid 202 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  Posetcpo 16861  ccvr 34026
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-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-rab 2916  df-v 3188  df-sbc 3418  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-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-iota 5810  df-fun 5849  df-fv 5855  df-preset 16849  df-poset 16867  df-plt 16879  df-covers 34030
This theorem is referenced by:  cvrcmp2  34048  atcmp  34075  llncmp  34285  lplncmp  34325  lvolcmp  34380  lhp2lt  34764
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