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Theorem pltirr 17278
Description: The "less than" relation is not reflexive. (pssirr 3904 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 2799 . 2 𝑋 = 𝑋
2 pltne.s . . . . 5 < = (lt‘𝐾)
32pltne 17277 . . . 4 ((𝐾𝐴𝑋𝐵𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
433anidm23 1545 . . 3 ((𝐾𝐴𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
54necon2bd 2987 . 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 385   = wceq 1653  wcel 2157  wne 2971   class class class wbr 4843  cfv 6101  ltcplt 17256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-plt 17273
This theorem is referenced by:  pospo  17288  atnlt  35334  llnnlt  35544  lplnnlt  35586  lvolnltN  35639
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