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Theorem pltnle 17187
Description: Less-than implies not inverse less-than-or-equal. (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 17181 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
4 pleval2.b . . . . . . . 8 𝐵 = (Base‘𝐾)
54, 1posasymb 17173 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
65biimpd 219 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
76expdimp 452 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑌 𝑋𝑋 = 𝑌))
87necon3ad 2945 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋𝑌 → ¬ 𝑌 𝑋))
98expimpd 630 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑋𝑌) → ¬ 𝑌 𝑋))
103, 9sylbid 230 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 𝑋))
1110imp 444 1 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  Basecbs 16079  lecple 16170  Posetcpo 17161  ltcplt 17162
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-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
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-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 17149  df-poset 17167  df-plt 17179
This theorem is referenced by:  pltnlt  17189  pltn2lp  17190  ncvr1  35080  cvrnle  35088  atlrelat1  35129  cvrat  35229
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