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Theorem ltltncvr 37932
Description: A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)
Hypotheses
Ref Expression
ltltncvr.b 𝐵 = (Base‘𝐾)
ltltncvr.s < = (lt‘𝐾)
ltltncvr.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
ltltncvr ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))

Proof of Theorem ltltncvr
StepHypRef Expression
1 simpll 766 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝐾𝐴)
2 simplr1 1216 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑋𝐵)
3 simplr3 1218 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑍𝐵)
4 simplr2 1217 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑌𝐵)
5 simpr 486 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑋𝐶𝑍)
6 ltltncvr.b . . . . 5 𝐵 = (Base‘𝐾)
7 ltltncvr.s . . . . 5 < = (lt‘𝐾)
8 ltltncvr.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
96, 7, 8cvrnbtwn 37779 . . . 4 ((𝐾𝐴 ∧ (𝑋𝐵𝑍𝐵𝑌𝐵) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
101, 2, 3, 4, 5, 9syl131anc 1384 . . 3 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
1110ex 414 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 → ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1211con2d 134 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107   class class class wbr 5106  cfv 6497  Basecbs 17088  ltcplt 18202  ccvr 37770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-covers 37774
This theorem is referenced by:  ltcvrntr  37933
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