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Theorem oplecon1b 37222
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 29872 analog.) (Contributed by NM, 6-Nov-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐵 = (Base‘𝐾)
opcon3.l = (le‘𝐾)
opcon3.o = (oc‘𝐾)
Assertion
Ref Expression
oplecon1b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))

Proof of Theorem oplecon1b
StepHypRef Expression
1 opcon3.b . . . . 5 𝐵 = (Base‘𝐾)
2 opcon3.o . . . . 5 = (oc‘𝐾)
31, 2opoccl 37215 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
433adant3 1131 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
5 opcon3.l . . . 4 = (le‘𝐾)
61, 5, 2oplecon3b 37221 . . 3 ((𝐾 ∈ OP ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) ( ‘( 𝑋))))
74, 6syld3an2 1410 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) ( ‘( 𝑋))))
81, 2opococ 37216 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
983adant3 1131 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
109breq2d 5087 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( ‘( 𝑋)) ↔ ( 𝑌) 𝑋))
117, 10bitrd 278 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1086   = wceq 1539  wcel 2107   class class class wbr 5075  cfv 6437  Basecbs 16921  lecple 16978  occoc 16979  OPcops 37193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-nul 5231
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-dm 5600  df-iota 6395  df-fv 6445  df-ov 7287  df-oposet 37197
This theorem is referenced by:  opoc1  37223  oldmm1  37238
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