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Theorem atncvrN 30127
Description: Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atncvr.c  |-  C  =  (  <o  `  K )
atncvr.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atncvrN  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P C Q )

Proof of Theorem atncvrN
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
2 atncvr.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atn0 30120 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  ( 0. `  K
) )
433adant3 975 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  P  =/=  ( 0. `  K
) )
5 eqid 2296 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
65, 2atbase 30101 . . . 4  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
7 eqid 2296 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
8 atncvr.c . . . . 5  |-  C  =  (  <o  `  K )
95, 7, 1, 8, 2atcvreq0 30126 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  ( Base `  K
)  /\  Q  e.  A )  ->  ( P C Q  <->  P  =  ( 0. `  K ) ) )
106, 9syl3an2 1216 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P C Q  <->  P  =  ( 0. `  K ) ) )
1110necon3bbid 2493 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( -.  P C Q  <->  P  =/=  ( 0. `  K ) ) )
124, 11mpbird 223 1  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P C Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-glb 14125  df-p0 14161  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110
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