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Theorem oplecon3b 38667
Description: Contraposition law for orthoposets. (chsscon3 31304 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐵 = (Base‘𝐾)
opcon3.l = (le‘𝐾)
opcon3.o = (oc‘𝐾)
Assertion
Ref Expression
oplecon3b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))

Proof of Theorem oplecon3b
StepHypRef Expression
1 opcon3.b . . 3 𝐵 = (Base‘𝐾)
2 opcon3.l . . 3 = (le‘𝐾)
3 opcon3.o . . 3 = (oc‘𝐾)
41, 2, 3oplecon3 38666 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))
5 simp1 1134 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
61, 3opoccl 38661 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
763adant2 1129 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
81, 3opoccl 38661 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
983adant3 1130 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
101, 2, 3oplecon3 38666 . . . 4 ((𝐾 ∈ OP ∧ ( 𝑌) ∈ 𝐵 ∧ ( 𝑋) ∈ 𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
115, 7, 9, 10syl3anc 1369 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
121, 3opococ 38662 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
13123adant3 1130 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
141, 3opococ 38662 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
15143adant2 1129 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1613, 15breq12d 5156 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) ( ‘( 𝑌)) ↔ 𝑋 𝑌))
1711, 16sylibd 238 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → 𝑋 𝑌))
184, 17impbid 211 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1085   = wceq 1534  wcel 2099   class class class wbr 5143  cfv 6543  Basecbs 17174  lecple 17234  occoc 17235  OPcops 38639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5301
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-dm 5683  df-iota 6495  df-fv 6551  df-ov 7418  df-oposet 38643
This theorem is referenced by:  oplecon1b  38668  opltcon3b  38671  oldmm1  38684  omllaw4  38713  cvrcmp2  38751  glbconN  38844  glbconNOLD  38845  lhpmod2i2  39506  lhpmod6i1  39507  lhprelat3N  39508  dochss  40833
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