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Theorem pltirr 17010
 Description: The less-than relation is not reflexive. (pssirr 3740 analog.) (Contributed by NM, 7-Feb-2012.)
Hypothesis
Ref Expression
pltne.s < = (lt‘𝐾)
Assertion
Ref Expression
pltirr ((𝐾𝐴𝑋𝐵) → ¬ 𝑋 < 𝑋)

Proof of Theorem pltirr
StepHypRef Expression
1 eqid 2651 . 2 𝑋 = 𝑋
2 pltne.s . . . . 5 < = (lt‘𝐾)
32pltne 17009 . . . 4 ((𝐾𝐴𝑋𝐵𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
433anidm23 1425 . . 3 ((𝐾𝐴𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
54necon2bd 2839 . 2 ((𝐾𝐴𝑋𝐵) → (𝑋 = 𝑋 → ¬ 𝑋 < 𝑋))
61, 5mpi 20 1 ((𝐾𝐴𝑋𝐵) → ¬ 𝑋 < 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823   class class class wbr 4685  ‘cfv 5926  ltcplt 16988 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-plt 17005 This theorem is referenced by:  pospo  17020  atnlt  34918  llnnlt  35127  lplnnlt  35169  lvolnltN  35222
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