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Theorem oplecon1b 36838
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 29436 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 36831 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
433adant3 1133 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
5 opcon3.l . . . 4 = (le‘𝐾)
61, 5, 2oplecon3b 36837 . . 3 ((𝐾 ∈ OP ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) ( ‘( 𝑋))))
74, 6syld3an2 1412 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) ( ‘( 𝑋))))
81, 2opococ 36832 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
983adant3 1133 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
109breq2d 5042 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( ‘( 𝑋)) ↔ ( 𝑌) 𝑋))
117, 10bitrd 282 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1088   = wceq 1542  wcel 2114   class class class wbr 5030  cfv 6339  Basecbs 16586  lecple 16675  occoc 16676  OPcops 36809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-dm 5535  df-iota 6297  df-fv 6347  df-ov 7173  df-oposet 36813
This theorem is referenced by:  opoc1  36839  oldmm1  36854
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