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Theorem leat2 29460
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 29458 . . . . 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 2627 . . . . . 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 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   0.cp0 14386   OPcops 29338   Atomscatm 29429
This theorem is referenced by:  dalemcea  29825  cdlemg12g  30814
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-glb 14352  df-p0 14388  df-oposet 29342  df-covers 29432  df-ats 29433
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