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Theorem oplecon1b 34307
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 28330 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 34300 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
433adant3 1079 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
5 opcon3.l . . . 4 = (le‘𝐾)
61, 5, 2oplecon3b 34306 . . 3 ((𝐾 ∈ OP ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) ( ‘( 𝑋))))
74, 6syld3an2 1371 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) ( ‘( 𝑋))))
81, 2opococ 34301 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
983adant3 1079 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
109breq2d 4656 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( ‘( 𝑋)) ↔ ( 𝑌) 𝑋))
117, 10bitrd 268 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1481  wcel 1988   class class class wbr 4644  cfv 5876  Basecbs 15838  lecple 15929  occoc 15930  OPcops 34278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-dm 5114  df-iota 5839  df-fv 5884  df-ov 6638  df-oposet 34282
This theorem is referenced by:  opoc1  34308  oldmm1  34323
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