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Theorem cvrnbtwn2 35065
Description: The covers relation implies no in-betweenness. (cvnbtwn2 29455 analog.) (Contributed by NM, 17-Nov-2011.)
Hypotheses
Ref Expression
cvrletr.b 𝐵 = (Base‘𝐾)
cvrletr.l = (le‘𝐾)
cvrletr.s < = (lt‘𝐾)
cvrletr.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrnbtwn2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))

Proof of Theorem cvrnbtwn2
StepHypRef Expression
1 cvrletr.b . . . . . 6 𝐵 = (Base‘𝐾)
2 cvrletr.s . . . . . 6 < = (lt‘𝐾)
3 cvrletr.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrnbtwn 35061 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
543expia 1115 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
6 iman 439 . . . . 5 (((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌))
7 simpl 474 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Poset)
8 simpr3 1238 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
9 simpr2 1236 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
10 cvrletr.l . . . . . . . . . . 11 = (le‘𝐾)
1110, 2pltval 17161 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
127, 8, 9, 11syl3anc 1477 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
13 df-ne 2933 . . . . . . . . . 10 (𝑍𝑌 ↔ ¬ 𝑍 = 𝑌)
1413anbi2i 732 . . . . . . . . 9 ((𝑍 𝑌𝑍𝑌) ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))
1512, 14syl6bb 276 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
1615anbi2d 742 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))))
17 anass 684 . . . . . . 7 (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
1816, 17syl6rbbr 279 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍𝑍 < 𝑌)))
1918notbid 307 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
206, 19syl5rbb 273 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
215, 20sylibd 229 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
22213impia 1110 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌))
231, 2, 3cvrlt 35060 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
2423ex 449 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 < 𝑌))
25243adant3r3 1200 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 < 𝑌))
26253impia 1110 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
27 breq2 4808 . . . 4 (𝑍 = 𝑌 → (𝑋 < 𝑍𝑋 < 𝑌))
2826, 27syl5ibrcom 237 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 < 𝑍))
291, 10posref 17152 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑌𝐵) → 𝑌 𝑌)
30293ad2antr2 1205 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌 𝑌)
31 breq1 4807 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑌𝑌 𝑌))
3230, 31syl5ibrcom 237 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 = 𝑌𝑍 𝑌))
33323adant3 1127 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3428, 33jcad 556 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → (𝑋 < 𝑍𝑍 𝑌)))
3522, 34impbid 202 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  Basecbs 16059  lecple 16150  Posetcpo 17141  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-ne 2933  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-preset 17129  df-poset 17147  df-plt 17159  df-covers 35056
This theorem is referenced by:  cvrval3  35202  cvrexchlem  35208
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