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Theorem oplecon1b 38157
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 30792 analog.) (Contributed by NM, 6-Nov-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐡 = (Baseβ€˜πΎ)
opcon3.l ≀ = (leβ€˜πΎ)
opcon3.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
oplecon1b ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ≀ π‘Œ ↔ ( βŠ₯ β€˜π‘Œ) ≀ 𝑋))

Proof of Theorem oplecon1b
StepHypRef Expression
1 opcon3.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 opcon3.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
31, 2opoccl 38150 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
433adant3 1132 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
5 opcon3.l . . . 4 ≀ = (leβ€˜πΎ)
61, 5, 2oplecon3b 38156 . . 3 ((𝐾 ∈ OP ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ≀ π‘Œ ↔ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))))
74, 6syld3an2 1411 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ≀ π‘Œ ↔ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))))
81, 2opococ 38151 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
983adant3 1132 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
109breq2d 5160 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) ↔ ( βŠ₯ β€˜π‘Œ) ≀ 𝑋))
117, 10bitrd 278 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ≀ π‘Œ ↔ ( βŠ₯ β€˜π‘Œ) ≀ 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  Basecbs 17146  lecple 17206  occoc 17207  OPcops 38128
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-dm 5686  df-iota 6495  df-fv 6551  df-ov 7414  df-oposet 38132
This theorem is referenced by:  opoc1  38158  oldmm1  38173
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