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Theorem pltirr 14420
Description: The less-than relation is not reflexive. (pssirr 3447 analog.) (Contributed by NM, 7-Feb-2012.)
Hypothesis
Ref Expression
pltne.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltirr  |-  ( ( K  e.  A  /\  X  e.  B )  ->  -.  X  .<  X )

Proof of Theorem pltirr
StepHypRef Expression
1 eqid 2436 . 2  |-  X  =  X
2 pltne.s . . . . 5  |-  .<  =  ( lt `  K )
32pltne 14419 . . . 4  |-  ( ( K  e.  A  /\  X  e.  B  /\  X  e.  B )  ->  ( X  .<  X  ->  X  =/=  X ) )
433anidm23 1243 . . 3  |-  ( ( K  e.  A  /\  X  e.  B )  ->  ( X  .<  X  ->  X  =/=  X ) )
54necon2bd 2653 . 2  |-  ( ( K  e.  A  /\  X  e.  B )  ->  ( X  =  X  ->  -.  X  .<  X ) )
61, 5mpi 17 1  |-  ( ( K  e.  A  /\  X  e.  B )  ->  -.  X  .<  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454   ltcplt 14398
This theorem is referenced by:  pospo  14430  atnlt  30111  llnnlt  30320  lplnnlt  30362  lvolnltN  30415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-plt 14415
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