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Theorem ltltncvr 38806
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 764 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝐾𝐴)
2 simplr1 1212 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑋𝐵)
3 simplr3 1214 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑍𝐵)
4 simplr2 1213 . . . 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 38653 . . . 4 ((𝐾𝐴 ∧ (𝑋𝐵𝑍𝐵𝑌𝐵) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
101, 2, 3, 4, 5, 9syl131anc 1380 . . 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 1084   = wceq 1533  wcel 2098   class class class wbr 5141  cfv 6536  Basecbs 17150  ltcplt 18270  ccvr 38644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-covers 38648
This theorem is referenced by:  ltcvrntr  38807
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