Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atnlt Unicode version

Theorem atnlt 29503
Description: Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)
Hypotheses
Ref Expression
atnlt.s  |-  .<  =  ( lt `  K )
atnlt.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atnlt  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P  .<  Q )

Proof of Theorem atnlt
StepHypRef Expression
1 atnlt.s . . . . 5  |-  .<  =  ( lt `  K )
21pltirr 14097 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<  P )
323adant3 975 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P  .<  P )
4 breq2 4027 . . . 4  |-  ( P  =  Q  ->  ( P  .<  P  <->  P  .<  Q ) )
54notbid 285 . . 3  |-  ( P  =  Q  ->  ( -.  P  .<  P  <->  -.  P  .<  Q ) )
63, 5syl5ibcom 211 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  -.  P  .<  Q )
)
7 eqid 2283 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
87, 1pltle 14095 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  Q  ->  P
( le `  K
) Q ) )
9 atnlt.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 9atcmp 29501 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P ( le `  K ) Q  <->  P  =  Q ) )
118, 10sylibd 205 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  Q  ->  P  =  Q ) )
1211necon3ad 2482 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  P  .<  Q ) )
136, 12pm2.61dne 2523 1  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P  .<  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   lecple 13215   ltcplt 14075   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  atltcvr  29624  llnnleat  29702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488
  Copyright terms: Public domain W3C validator