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Theorem opnlen0 36939
Description: An element not less than another is nonzero. TODO: Look for uses of necon3bd 2954 and op0le 36937 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 ∈ OP ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑋 𝑌) → 𝑋0 )

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 36937 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → 0 𝑌)
543adant2 1133 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → 0 𝑌)
6 breq1 5056 . . . 4 (𝑋 = 0 → (𝑋 𝑌0 𝑌))
75, 6syl5ibrcom 250 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 0𝑋 𝑌))
87necon3bd 2954 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌𝑋0 ))
98imp 410 1 (((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑋 𝑌) → 𝑋0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940   class class class wbr 5053  cfv 6380  Basecbs 16760  lecple 16809  0.cp0 17929  OPcops 36923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-glb 17853  df-p0 17931  df-oposet 36927
This theorem is referenced by:  cdlemg12e  38398
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