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Theorem leat2 29777
Description: A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
Hypotheses
Ref Expression
leatom.b  |-  B  =  ( Base `  K
)
leatom.l  |-  .<_  =  ( le `  K )
leatom.z  |-  .0.  =  ( 0. `  K )
leatom.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
leat2  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  ( X  =/=  .0.  /\  X  .<_  P )
)  ->  X  =  P )

Proof of Theorem leat2
StepHypRef Expression
1 leatom.b . . . . . 6  |-  B  =  ( Base `  K
)
2 leatom.l . . . . . 6  |-  .<_  =  ( le `  K )
3 leatom.z . . . . . 6  |-  .0.  =  ( 0. `  K )
4 leatom.a . . . . . 6  |-  A  =  ( Atoms `  K )
51, 2, 3, 4leatb 29775 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  <->  ( X  =  P  \/  X  =  .0.  ) ) )
6 orcom 377 . . . . . 6  |-  ( ( X  =  P  \/  X  =  .0.  )  <->  ( X  =  .0.  \/  X  =  P )
)
7 neor 2651 . . . . . 6  |-  ( ( X  =  .0.  \/  X  =  P )  <->  ( X  =/=  .0.  ->  X  =  P ) )
86, 7bitri 241 . . . . 5  |-  ( ( X  =  P  \/  X  =  .0.  )  <->  ( X  =/=  .0.  ->  X  =  P ) )
95, 8syl6bb 253 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  <->  ( X  =/=  .0.  ->  X  =  P ) ) )
109biimpd 199 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  -> 
( X  =/=  .0.  ->  X  =  P ) ) )
1110com23 74 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  =/=  .0.  ->  ( X  .<_  P  ->  X  =  P )
) )
1211imp32 423 1  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  ( X  =/=  .0.  /\  X  .<_  P )
)  ->  X  =  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   0.cp0 14421   OPcops 29655   Atomscatm 29746
This theorem is referenced by:  dalemcea  30142  cdlemg12g  31131
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-rep 4280  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-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  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-iun 4055  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-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-glb 14387  df-p0 14423  df-oposet 29659  df-covers 29749  df-ats 29750
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