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Theorem opnoncon 36348
Description: Law of contradiction for orthoposets. (chocin 29275 analog.) (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opnoncon.b 𝐵 = (Base‘𝐾)
opnoncon.o = (oc‘𝐾)
opnoncon.m = (meet‘𝐾)
opnoncon.z 0 = (0.‘𝐾)
Assertion
Ref Expression
opnoncon ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 0 )

Proof of Theorem opnoncon
StepHypRef Expression
1 opnoncon.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2824 . . . 4 (le‘𝐾) = (le‘𝐾)
3 opnoncon.o . . . 4 = (oc‘𝐾)
4 eqid 2824 . . . 4 (join‘𝐾) = (join‘𝐾)
5 opnoncon.m . . . 4 = (meet‘𝐾)
6 opnoncon.z . . . 4 0 = (0.‘𝐾)
7 eqid 2824 . . . 4 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 36322 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑋𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))) ∧ (𝑋(join‘𝐾)( 𝑋)) = (1.‘𝐾) ∧ (𝑋 ( 𝑋)) = 0 ))
983anidm23 1417 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))) ∧ (𝑋(join‘𝐾)( 𝑋)) = (1.‘𝐾) ∧ (𝑋 ( 𝑋)) = 0 ))
109simp3d 1140 1 ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1536  wcel 2113   class class class wbr 5069  cfv 6358  (class class class)co 7159  Basecbs 16486  lecple 16575  occoc 16576  joincjn 17557  meetcmee 17558  0.cp0 17650  1.cp1 17651  OPcops 36312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-nul 5213
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-dm 5568  df-iota 6317  df-fv 6366  df-ov 7162  df-oposet 36316
This theorem is referenced by:  omlfh1N  36398  omlspjN  36401  atlatmstc  36459  pnonsingN  37073  lhpocnle  37156  dochnoncon  38531
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