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Theorem oplecon3b 36323
Description: Contraposition law for orthoposets. (chsscon3 29269 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 36322 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))
5 simp1 1130 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
61, 3opoccl 36317 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
763adant2 1125 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
81, 3opoccl 36317 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
983adant3 1126 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
101, 2, 3oplecon3 36322 . . . 4 ((𝐾 ∈ OP ∧ ( 𝑌) ∈ 𝐵 ∧ ( 𝑋) ∈ 𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
115, 7, 9, 10syl3anc 1365 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
121, 3opococ 36318 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
13123adant3 1126 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
141, 3opococ 36318 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
15143adant2 1125 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1613, 15breq12d 5070 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) ( ‘( 𝑌)) ↔ 𝑋 𝑌))
1711, 16sylibd 241 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → 𝑋 𝑌))
184, 17impbid 214 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5057  cfv 6348  Basecbs 16475  lecple 16564  occoc 16565  OPcops 36295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7151  df-oposet 36299
This theorem is referenced by:  oplecon1b  36324  opltcon3b  36327  oldmm1  36340  omllaw4  36369  cvrcmp2  36407  glbconN  36500  lhpmod2i2  37161  lhpmod6i1  37162  lhprelat3N  37163  dochss  38488
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