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Theorem atnlt 29796
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 14375 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<  P )
323adant3 977 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P  .<  P )
4 breq2 4176 . . . 4  |-  ( P  =  Q  ->  ( P  .<  P  <->  P  .<  Q ) )
54notbid 286 . . 3  |-  ( P  =  Q  ->  ( -.  P  .<  P  <->  -.  P  .<  Q ) )
63, 5syl5ibcom 212 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  -.  P  .<  Q )
)
7 eqid 2404 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
87, 1pltle 14373 . . . 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 29794 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P ( le `  K ) Q  <->  P  =  Q ) )
118, 10sylibd 206 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  Q  ->  P  =  Q ) )
1211necon3ad 2603 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  P  .<  Q ) )
136, 12pm2.61dne 2644 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 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413   lecple 13491   ltcplt 14353   Atomscatm 29746   AtLatcal 29747
This theorem is referenced by:  atltcvr  29917  llnnleat  29995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-poset 14358  df-plt 14370  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781
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