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Theorem opnoncon 36224
Description: Law of contradiction for orthoposets. (chocin 29199 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 2818 . . . 4 (le‘𝐾) = (le‘𝐾)
3 opnoncon.o . . . 4 = (oc‘𝐾)
4 eqid 2818 . . . 4 (join‘𝐾) = (join‘𝐾)
5 opnoncon.m . . . 4 = (meet‘𝐾)
6 opnoncon.z . . . 4 0 = (0.‘𝐾)
7 eqid 2818 . . . 4 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 36198 . . 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 1413 . 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 1136 1 ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  occoc 16561  joincjn 17542  meetcmee 17543  0.cp0 17635  1.cp1 17636  OPcops 36188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7148  df-oposet 36192
This theorem is referenced by:  omlfh1N  36274  omlspjN  36277  atlatmstc  36335  pnonsingN  36949  lhpocnle  37032  dochnoncon  38407
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