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Theorem ltltncvr 39405
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 767 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝐾𝐴)
2 simplr1 1214 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑋𝐵)
3 simplr3 1216 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑍𝐵)
4 simplr2 1215 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑌𝐵)
5 simpr 484 . . . 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 39252 . . . 4 ((𝐾𝐴 ∧ (𝑋𝐵𝑍𝐵𝑌𝐵) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
101, 2, 3, 4, 5, 9syl131anc 1382 . . 3 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
1110ex 412 . 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 395  w3a 1086   = wceq 1536  wcel 2105   class class class wbr 5147  cfv 6562  Basecbs 17244  ltcplt 18365  ccvr 39243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-covers 39247
This theorem is referenced by:  ltcvrntr  39406
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