Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opnoncon Structured version   Visualization version   GIF version

Theorem opnoncon 37673
Description: Law of contradiction for orthoposets. (chocin 30440 analog.) (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opnoncon.b 𝐡 = (Baseβ€˜πΎ)
opnoncon.o βŠ₯ = (ocβ€˜πΎ)
opnoncon.m ∧ = (meetβ€˜πΎ)
opnoncon.z 0 = (0.β€˜πΎ)
Assertion
Ref Expression
opnoncon ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 )

Proof of Theorem opnoncon
StepHypRef Expression
1 opnoncon.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 eqid 2737 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
3 opnoncon.o . . . 4 βŠ₯ = (ocβ€˜πΎ)
4 eqid 2737 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
5 opnoncon.m . . . 4 ∧ = (meetβ€˜πΎ)
6 opnoncon.z . . . 4 0 = (0.β€˜πΎ)
7 eqid 2737 . . . 4 (1.β€˜πΎ) = (1.β€˜πΎ)
81, 2, 3, 4, 5, 6, 7oposlem 37647 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋(leβ€˜πΎ)𝑋 β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹))) ∧ (𝑋(joinβ€˜πΎ)( βŠ₯ β€˜π‘‹)) = (1.β€˜πΎ) ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 ))
983anidm23 1422 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋(leβ€˜πΎ)𝑋 β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹))) ∧ (𝑋(joinβ€˜πΎ)( βŠ₯ β€˜π‘‹)) = (1.β€˜πΎ) ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 ))
109simp3d 1145 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  occoc 17142  joincjn 18201  meetcmee 18202  0.cp0 18313  1.cp1 18314  OPcops 37637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-ov 7361  df-oposet 37641
This theorem is referenced by:  omlfh1N  37723  omlspjN  37726  atlatmstc  37784  pnonsingN  38399  lhpocnle  38482  dochnoncon  39857
  Copyright terms: Public domain W3C validator