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Theorem atnlt 30111
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 14420 . . . 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 4216 . . . 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 2436 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
87, 1pltle 14418 . . . 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 30109 . . . 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 2637 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  P  .<  Q ) )
136, 12pm2.61dne 2681 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 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   lecple 13536   ltcplt 14398   Atomscatm 30061   AtLatcal 30062
This theorem is referenced by:  atltcvr  30232  llnnleat  30310
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-13 1727  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-pow 4377  ax-pr 4403  ax-un 4701
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-pw 3801  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-ov 6084  df-poset 14403  df-plt 14415  df-lat 14475  df-covers 30064  df-ats 30065  df-atl 30096
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