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Theorem ltltncvr 35212
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 807 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝐾𝐴)
2 simplr1 1261 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑋𝐵)
3 simplr3 1265 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑍𝐵)
4 simplr2 1263 . . . 4 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → 𝑌𝐵)
5 simpr 479 . . . 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 35061 . . . 4 ((𝐾𝐴 ∧ (𝑋𝐵𝑍𝐵𝑌𝐵) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
101, 2, 3, 4, 5, 9syl131anc 1490 . . 3 (((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → ¬ (𝑋 < 𝑌𝑌 < 𝑍))
1110ex 449 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 → ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1211con2d 129 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  Basecbs 16059  ltcplt 17142  ccvr 35052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-covers 35056
This theorem is referenced by:  ltcvrntr  35213
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