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Theorem opnlen0 29378
Description: An element not less than another is nonzero. TODO: Look for uses of necon3bd 2483 and op0le 29376 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 29376 . . . . 5  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  .0.  .<_  Y )
543adant2 974 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  .0.  .<_  Y )
6 breq1 4026 . . . 4  |-  ( X  =  .0.  ->  ( X  .<_  Y  <->  .0.  .<_  Y ) )
75, 6syl5ibrcom 213 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  .0. 
->  X  .<_  Y ) )
87necon3bd 2483 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  ->  X  =/=  .0.  ) )
98imp 418 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   OPcops 29362
This theorem is referenced by:  cdlemg12e  30836
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-glb 14109  df-p0 14145  df-oposet 29366
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