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Theorem cvrcmp 35085
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 1226 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝐾 ∈ Poset)
2 simpl23 1323 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐵)
3 simpl21 1319 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋𝐵)
4 simpl3l 1285 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑋)
5 cvrcmp.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cvrcmp.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
75, 6cvrne 35083 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍𝑋)
81, 2, 3, 4, 7syl31anc 1478 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝑋)
9 cvrcmp.l . . . . . . . 8 = (le‘𝐾)
105, 9, 6cvrle 35080 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 𝑋)
111, 2, 3, 4, 10syl31anc 1478 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍 𝑋)
12 simpr 471 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 𝑌)
13 simpl22 1321 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑌𝐵)
14 simpl3r 1287 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑌)
155, 9, 6cvrnbtwn4 35081 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑍𝐵𝑌𝐵𝑋𝐵) ∧ 𝑍𝐶𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
161, 2, 13, 3, 14, 15syl131anc 1488 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
1711, 12, 16mpbi2and 683 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍 = 𝑋𝑋 = 𝑌))
18 neor 3033 . . . . 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 397 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
22 simp1 1129 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23 simp21 1247 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋𝐵)
245, 9posref 17158 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
2522, 23, 24syl2anc 565 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋 𝑋)
26 breq2 4788 . . 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  wa 382  wo 826  w3a 1070   = wceq 1630  wcel 2144  wne 2942   class class class wbr 4784  cfv 6031  Basecbs 16063  lecple 16155  Posetcpo 17147  ccvr 35064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-preset 17135  df-poset 17153  df-plt 17165  df-covers 35068
This theorem is referenced by:  cvrcmp2  35086  atcmp  35113  llncmp  35323  lplncmp  35363  lvolcmp  35418  lhp2lt  35802
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