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Theorem opnoncon 38066
Description: Law of contradiction for orthoposets. (chocin 30735 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 2732 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
3 opnoncon.o . . . 4 βŠ₯ = (ocβ€˜πΎ)
4 eqid 2732 . . . 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 2732 . . . 4 (1.β€˜πΎ) = (1.β€˜πΎ)
81, 2, 3, 4, 5, 6, 7oposlem 38040 . . 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 1421 . 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 1144 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  occoc 17201  joincjn 18260  meetcmee 18261  0.cp0 18372  1.cp1 18373  OPcops 38030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-dm 5685  df-iota 6492  df-fv 6548  df-ov 7408  df-oposet 38034
This theorem is referenced by:  omlfh1N  38116  omlspjN  38119  atlatmstc  38177  pnonsingN  38792  lhpocnle  38875  dochnoncon  40250
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