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Theorem opnlen0 29683
Description: An element not less than another is nonzero. TODO: Look for uses of necon3bd 2612 and op0le 29681 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
Hypotheses
Ref Expression
op0le.b  |-  B  =  ( Base `  K
)
op0le.l  |-  .<_  =  ( le `  K )
op0le.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
opnlen0  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  /\  -.  X  .<_  Y )  ->  X  =/=  .0.  )

Proof of Theorem opnlen0
StepHypRef Expression
1 op0le.b . . . . . 6  |-  B  =  ( Base `  K
)
2 op0le.l . . . . . 6  |-  .<_  =  ( le `  K )
3 op0le.z . . . . . 6  |-  .0.  =  ( 0. `  K )
41, 2, 3op0le 29681 . . . . 5  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  .0.  .<_  Y )
543adant2 976 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  .0.  .<_  Y )
6 breq1 4183 . . . 4  |-  ( X  =  .0.  ->  ( X  .<_  Y  <->  .0.  .<_  Y ) )
75, 6syl5ibrcom 214 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  .0. 
->  X  .<_  Y ) )
87necon3bd 2612 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  ->  X  =/=  .0.  ) )
98imp 419 1  |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  /\  -.  X  .<_  Y )  ->  X  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180   ` cfv 5421   Basecbs 13432   lecple 13499   0.cp0 14429   OPcops 29667
This theorem is referenced by:  cdlemg12e  31141
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-undef 6510  df-riota 6516  df-glb 14395  df-p0 14431  df-oposet 29671
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