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Theorem pltnle 17576
Description: "Less than" implies not converse "less than or equal to". (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
pleval2.b 𝐵 = (Base‘𝐾)
pleval2.l = (le‘𝐾)
pleval2.s < = (lt‘𝐾)
Assertion
Ref Expression
pltnle (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 𝑋)

Proof of Theorem pltnle
StepHypRef Expression
1 pleval2.l . . . 4 = (le‘𝐾)
2 pleval2.s . . . 4 < = (lt‘𝐾)
31, 2pltval 17570 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
4 pleval2.b . . . . . . . 8 𝐵 = (Base‘𝐾)
54, 1posasymb 17562 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
65biimpd 232 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
76expdimp 456 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑌 𝑋𝑋 = 𝑌))
87necon3ad 3027 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋𝑌 → ¬ 𝑌 𝑋))
98expimpd 457 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑋𝑌) → ¬ 𝑌 𝑋))
103, 9sylbid 243 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 𝑋))
1110imp 410 1 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  Basecbs 16483  lecple 16572  Posetcpo 17550  ltcplt 17551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-proset 17538  df-poset 17556  df-plt 17568
This theorem is referenced by:  pltnlt  17578  pltn2lp  17579  ncvr1  36516  cvrnle  36524  atlrelat1  36565  cvrat  36666
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