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Theorem opnlen0 29987
Description: An element not less than another is nonzero. TODO: Look for uses of necon3bd 2639 and op0le 29985 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 29985 . . . . 5  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  .0.  .<_  Y )
543adant2 977 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  .0.  .<_  Y )
6 breq1 4216 . . . 4  |-  ( X  =  .0.  ->  ( X  .<_  Y  <->  .0.  .<_  Y ) )
75, 6syl5ibrcom 215 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  .0. 
->  X  .<_  Y ) )
87necon3bd 2639 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  ->  X  =/=  .0.  ) )
98imp 420 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   class class class wbr 4213   ` cfv 5455   Basecbs 13470   lecple 13537   0.cp0 14467   OPcops 29971
This theorem is referenced by:  cdlemg12e  31445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-undef 6544  df-riota 6550  df-glb 14433  df-p0 14469  df-oposet 29975
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