Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opcon3b Structured version   Visualization version   GIF version

Theorem opcon3b 39178
Description: Contraposition law for orthoposets. (chcon3i 31495 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
opoccl.b 𝐵 = (Base‘𝐾)
opoccl.o = (oc‘𝐾)
Assertion
Ref Expression
opcon3b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))

Proof of Theorem opcon3b
StepHypRef Expression
1 fveq2 6907 . . 3 (𝑌 = 𝑋 → ( 𝑌) = ( 𝑋))
21eqcoms 2743 . 2 (𝑋 = 𝑌 → ( 𝑌) = ( 𝑋))
3 fveq2 6907 . . . 4 (( 𝑋) = ( 𝑌) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
43eqcoms 2743 . . 3 (( 𝑌) = ( 𝑋) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
5 opoccl.b . . . . . 6 𝐵 = (Base‘𝐾)
6 opoccl.o . . . . . 6 = (oc‘𝐾)
75, 6opococ 39177 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
873adant3 1131 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
95, 6opococ 39177 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1093adant2 1130 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
118, 10eqeq12d 2751 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) = ( ‘( 𝑌)) ↔ 𝑋 = 𝑌))
124, 11imbitrid 244 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) = ( 𝑋) → 𝑋 = 𝑌))
132, 12impbid2 226 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  Basecbs 17245  occoc 17306  OPcops 39154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-oposet 39158
This theorem is referenced by:  opcon2b  39179  omllaw4  39228  cmtbr2N  39235  cvrcmp2  39266  lhpmod2i2  40021  lhpmod6i1  40022
  Copyright terms: Public domain W3C validator