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Theorem opcon2b 36398
Description: Orthocomplement contraposition law. (negcon2 10926 analog.) (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
opoccl.b 𝐵 = (Base‘𝐾)
opoccl.o = (oc‘𝐾)
Assertion
Ref Expression
opcon2b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))

Proof of Theorem opcon2b
StepHypRef Expression
1 opoccl.b . . . . 5 𝐵 = (Base‘𝐾)
2 opoccl.o . . . . 5 = (oc‘𝐾)
31, 2opoccl 36395 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
433adant2 1128 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
51, 2opcon3b 36397 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 = ( 𝑌) ↔ ( ‘( 𝑌)) = ( 𝑋)))
64, 5syld3an3 1406 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ ( ‘( 𝑌)) = ( 𝑋)))
71, 2opococ 36396 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
873adant2 1128 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
98eqeq1d 2826 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑌)) = ( 𝑋) ↔ 𝑌 = ( 𝑋)))
106, 9bitrd 282 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2115  cfv 6338  Basecbs 16474  occoc 16564  OPcops 36373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5193
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-dm 5548  df-iota 6297  df-fv 6346  df-ov 7143  df-oposet 36377
This theorem is referenced by:  opcon1b  36399  riotaocN  36410  glbconxN  36579
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