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Theorem oplecon3 39181
Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐵 = (Base‘𝐾)
opcon3.l = (le‘𝐾)
opcon3.o = (oc‘𝐾)
Assertion
Ref Expression
oplecon3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))

Proof of Theorem oplecon3
StepHypRef Expression
1 opcon3.b . . . 4 𝐵 = (Base‘𝐾)
2 opcon3.l . . . 4 = (le‘𝐾)
3 opcon3.o . . . 4 = (oc‘𝐾)
4 eqid 2735 . . . 4 (join‘𝐾) = (join‘𝐾)
5 eqid 2735 . . . 4 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2735 . . . 4 (0.‘𝐾) = (0.‘𝐾)
7 eqid 2735 . . . 4 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 39164 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋(join‘𝐾)( 𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( 𝑋)) = (0.‘𝐾)))
98simp1d 1141 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))))
109simp3d 1143 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  occoc 17306  joincjn 18369  meetcmee 18370  0.cp0 18481  1.cp1 18482  OPcops 39154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-oposet 39158
This theorem is referenced by:  oplecon3b  39182
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