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Theorem opltcon3b 38587
Description: Contraposition law for strict ordering in orthoposets. (chpsscon3 31265 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
opltcon3.b 𝐡 = (Baseβ€˜πΎ)
opltcon3.s < = (ltβ€˜πΎ)
opltcon3.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
opltcon3b ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ ↔ ( βŠ₯ β€˜π‘Œ) < ( βŠ₯ β€˜π‘‹)))

Proof of Theorem opltcon3b
StepHypRef Expression
1 opltcon3.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 eqid 2726 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
3 opltcon3.o . . . 4 βŠ₯ = (ocβ€˜πΎ)
41, 2, 3oplecon3b 38583 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(leβ€˜πΎ)π‘Œ ↔ ( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹)))
51, 2, 3oplecon3b 38583 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (π‘Œ(leβ€˜πΎ)𝑋 ↔ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘Œ)))
653com23 1123 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ(leβ€˜πΎ)𝑋 ↔ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘Œ)))
76notbid 318 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (Β¬ π‘Œ(leβ€˜πΎ)𝑋 ↔ Β¬ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘Œ)))
84, 7anbi12d 630 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋(leβ€˜πΎ)π‘Œ ∧ Β¬ π‘Œ(leβ€˜πΎ)𝑋) ↔ (( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹) ∧ Β¬ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘Œ))))
9 opposet 38564 . . 3 (𝐾 ∈ OP β†’ 𝐾 ∈ Poset)
10 opltcon3.s . . . 4 < = (ltβ€˜πΎ)
111, 2, 10pltval3 18304 . . 3 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ ↔ (𝑋(leβ€˜πΎ)π‘Œ ∧ Β¬ π‘Œ(leβ€˜πΎ)𝑋)))
129, 11syl3an1 1160 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ ↔ (𝑋(leβ€˜πΎ)π‘Œ ∧ Β¬ π‘Œ(leβ€˜πΎ)𝑋)))
1393ad2ant1 1130 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Poset)
141, 3opoccl 38577 . . . 4 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
15143adant2 1128 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
161, 3opoccl 38577 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
17163adant3 1129 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
181, 2, 10pltval3 18304 . . 3 ((𝐾 ∈ Poset ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘Œ) < ( βŠ₯ β€˜π‘‹) ↔ (( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹) ∧ Β¬ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘Œ))))
1913, 15, 17, 18syl3anc 1368 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘Œ) < ( βŠ₯ β€˜π‘‹) ↔ (( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜π‘‹) ∧ Β¬ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜π‘Œ))))
208, 12, 193bitr4d 311 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ ↔ ( βŠ₯ β€˜π‘Œ) < ( βŠ₯ β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6537  Basecbs 17153  lecple 17213  occoc 17214  Posetcpo 18272  ltcplt 18273  OPcops 38555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-proset 18260  df-poset 18278  df-plt 18295  df-oposet 38559
This theorem is referenced by:  opltcon1b  38588  opltcon2b  38589  cvrcon3b  38660  1cvratex  38857  lhprelat3N  39424
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