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Theorem opnlen0 34997
Description: An element not less than another is nonzero. TODO: Look for uses of necon3bd 2957 and op0le 34995 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 34995 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → 0 𝑌)
543adant2 1125 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → 0 𝑌)
6 breq1 4790 . . . 4 (𝑋 = 0 → (𝑋 𝑌0 𝑌))
75, 6syl5ibrcom 237 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 0𝑋 𝑌))
87necon3bd 2957 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌𝑋0 ))
98imp 393 1 (((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑋 𝑌) → 𝑋0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4787  cfv 6030  Basecbs 16064  lecple 16156  0.cp0 17245  OPcops 34981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-glb 17183  df-p0 17247  df-oposet 34985
This theorem is referenced by:  cdlemg12e  36457
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