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Theorem leat3 29303
Description: A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
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
leat3  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( X  e.  A  \/  X  =  .0.  ) )

Proof of Theorem leat3
StepHypRef Expression
1 leatom.b . . 3  |-  B  =  ( Base `  K
)
2 leatom.l . . 3  |-  .<_  =  ( le `  K )
3 leatom.z . . 3  |-  .0.  =  ( 0. `  K )
4 leatom.a . . 3  |-  A  =  ( Atoms `  K )
51, 2, 3, 4leat 29301 . 2  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( X  =  P  \/  X  =  .0.  ) )
6 simpl3 960 . . . 4  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  P  e.  A
)
7 eleq1a 2385 . . . 4  |-  ( P  e.  A  ->  ( X  =  P  ->  X  e.  A ) )
86, 7syl 15 . . 3  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( X  =  P  ->  X  e.  A ) )
98orim1d 812 . 2  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( ( X  =  P  \/  X  =  .0.  )  ->  ( X  e.  A  \/  X  =  .0.  )
) )
105, 9mpd 14 1  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( X  e.  A  \/  X  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292   Basecbs 13195   lecple 13262   0.cp0 14192   OPcops 29180   Atomscatm 29271
This theorem is referenced by:  cdleme22b  30348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-glb 14158  df-p0 14194  df-oposet 29184  df-covers 29274  df-ats 29275
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