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Theorem oplecon3b 33505
 Description: Contraposition law for orthoposets. (chsscon3 27743 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 33504 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))
5 simp1 1054 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
61, 3opoccl 33499 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
763adant2 1073 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
81, 3opoccl 33499 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
983adant3 1074 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
101, 2, 3oplecon3 33504 . . . 4 ((𝐾 ∈ OP ∧ ( 𝑌) ∈ 𝐵 ∧ ( 𝑋) ∈ 𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
115, 7, 9, 10syl3anc 1318 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
121, 3opococ 33500 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
13123adant3 1074 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
141, 3opococ 33500 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
15143adant2 1073 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1613, 15breq12d 4596 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) ( ‘( 𝑌)) ↔ 𝑋 𝑌))
1711, 16sylibd 228 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → 𝑋 𝑌))
184, 17impbid 201 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  occoc 15776  OPcops 33477 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-oposet 33481 This theorem is referenced by:  oplecon1b  33506  opltcon3b  33509  oldmm1  33522  omllaw4  33551  cvrcmp2  33589  glbconN  33681  lhpmod2i2  34342  lhpmod6i1  34343  lhprelat3N  34344  dochss  35672
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