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Theorem atncvrN 29505
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 2283 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
2 atncvr.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atn0 29498 . . 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 2283 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
65, 2atbase 29479 . . . 4  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
7 eqid 2283 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
8 atncvr.c . . . . 5  |-  C  =  (  <o  `  K )
95, 7, 1, 8, 2atcvreq0 29504 . . . 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 2480 . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454
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-rep 4131  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-glb 14109  df-p0 14145  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488
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