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Theorem oplecon3b 35221
Description: Contraposition law for orthoposets. (chsscon3 28884 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 35220 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))
5 simp1 1167 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
61, 3opoccl 35215 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
763adant2 1162 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
81, 3opoccl 35215 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
983adant3 1163 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
101, 2, 3oplecon3 35220 . . . 4 ((𝐾 ∈ OP ∧ ( 𝑌) ∈ 𝐵 ∧ ( 𝑋) ∈ 𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
115, 7, 9, 10syl3anc 1491 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
121, 3opococ 35216 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
13123adant3 1163 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
141, 3opococ 35216 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
15143adant2 1162 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1613, 15breq12d 4856 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) ( ‘( 𝑌)) ↔ 𝑋 𝑌))
1711, 16sylibd 231 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → 𝑋 𝑌))
184, 17impbid 204 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108   = wceq 1653  wcel 2157   class class class wbr 4843  cfv 6101  Basecbs 16184  lecple 16274  occoc 16275  OPcops 35193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-dm 5322  df-iota 6064  df-fv 6109  df-ov 6881  df-oposet 35197
This theorem is referenced by:  oplecon1b  35222  opltcon3b  35225  oldmm1  35238  omllaw4  35267  cvrcmp2  35305  glbconN  35398  lhpmod2i2  36059  lhpmod6i1  36060  lhprelat3N  36061  dochss  37386
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