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Theorem oplecon3 39201
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 2736 . . . 4 (join‘𝐾) = (join‘𝐾)
5 eqid 2736 . . . 4 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2736 . . . 4 (0.‘𝐾) = (0.‘𝐾)
7 eqid 2736 . . . 4 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 39184 . . 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 1142 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))))
109simp3d 1144 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2107   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  occoc 17306  joincjn 18358  meetcmee 18359  0.cp0 18469  1.cp1 18470  OPcops 39174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-dm 5694  df-iota 6513  df-fv 6568  df-ov 7435  df-oposet 39178
This theorem is referenced by:  oplecon3b  39202
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