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Theorem oplecon1b 37666
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 30446 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 37659 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
433adant3 1133 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
5 opcon3.l . . . 4 ≀ = (leβ€˜πΎ)
61, 5, 2oplecon3b 37665 . . 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 1412 . 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 37660 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
983adant3 1133 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
109breq2d 5118 . 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 279 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 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  Basecbs 17084  lecple 17141  occoc 17142  OPcops 37637
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-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-ov 7361  df-oposet 37641
This theorem is referenced by:  opoc1  37667  oldmm1  37682
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