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Theorem ltltncvr 33510
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 785 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝐾𝐴)
2 simplr1 1095 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑋𝐵)
3 simplr3 1097 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑍𝐵)
4 simplr2 1096 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑌𝐵)
5 simpr 475 . . . 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 33359 . . . 4 ((𝐾𝐴 ∧ (𝑋𝐵𝑍𝐵𝑌𝐵) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
101, 2, 3, 4, 5, 9syl131anc 1330 . . 3 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
1110ex 448 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 → ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1211con2d 127 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976   class class class wbr 4577  cfv 5789  Basecbs 15643  ltcplt 16712  ccvr 33350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-covers 33354
This theorem is referenced by:  ltcvrntr  33511
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