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Theorem oplecon3 38372
Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐡 = (Baseβ€˜πΎ)
opcon3.l ≀ = (leβ€˜πΎ)
opcon3.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
oplecon3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹)))

Proof of Theorem oplecon3
StepHypRef Expression
1 opcon3.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 opcon3.l . . . 4 ≀ = (leβ€˜πΎ)
3 opcon3.o . . . 4 βŠ₯ = (ocβ€˜πΎ)
4 eqid 2730 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
5 eqid 2730 . . . 4 (meetβ€˜πΎ) = (meetβ€˜πΎ)
6 eqid 2730 . . . 4 (0.β€˜πΎ) = (0.β€˜πΎ)
7 eqid 2730 . . . 4 (1.β€˜πΎ) = (1.β€˜πΎ)
81, 2, 3, 4, 5, 6, 7oposlem 38355 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹))) ∧ (𝑋(joinβ€˜πΎ)( βŠ₯ β€˜π‘‹)) = (1.β€˜πΎ) ∧ (𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘‹)) = (0.β€˜πΎ)))
98simp1d 1140 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹))))
109simp3d 1142 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  lecple 17208  occoc 17209  joincjn 18268  meetcmee 18269  0.cp0 18380  1.cp1 18381  OPcops 38345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-dm 5685  df-iota 6494  df-fv 6550  df-ov 7414  df-oposet 38349
This theorem is referenced by:  oplecon3b  38373
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