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Theorem pltirr 17690
Description: The "less than" relation is not reflexive. (pssirr 3992 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 2738 . 2 𝑋 = 𝑋
2 pltne.s . . . . 5 < = (lt‘𝐾)
32pltne 17689 . . . 4 ((𝐾𝐴𝑋𝐵𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
433anidm23 1422 . . 3 ((𝐾𝐴𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
54necon2bd 2950 . 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 399   = wceq 1542  wcel 2113  wne 2934   class class class wbr 5031  cfv 6340  ltcplt 17668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-iota 6298  df-fun 6342  df-fv 6348  df-plt 17685
This theorem is referenced by:  pospo  17700  atnlt  36947  llnnlt  37157  lplnnlt  37199  lvolnltN  37252
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