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Theorem outsideoftr 35405
Description: Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsideoftr ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ 𝑃OutsideOf⟨𝐡, 𝐢⟩) β†’ 𝑃OutsideOf⟨𝐴, 𝐢⟩))

Proof of Theorem outsideoftr
StepHypRef Expression
1 simpll 763 . . . . 5 (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) β†’ 𝐴 β‰  𝑃)
2 simplr 765 . . . . 5 (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) β†’ 𝐡 β‰  𝑃)
3 simprr 769 . . . . 5 (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) β†’ 𝐢 β‰  𝑃)
41, 2, 33jca 1126 . . . 4 (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) β†’ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃))
5 simplr1 1213 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝐴 β‰  𝑃)
6 simplr3 1215 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝐢 β‰  𝑃)
7 df-3an 1087 . . . . . . . . . . . 12 (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩) ↔ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩))
8 simp1 1134 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
9 simp3r 1200 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
10 simp2l 1197 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
11 simp2r 1198 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
12 simp3l 1199 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
13 simpr2 1193 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)
14 simpr3 1194 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)
158, 9, 10, 11, 12, 13, 14btwnexchand 35302 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩)
1615orcd 869 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))
177, 16sylan2br 593 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))
1817expr 455 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
19 simprlr 776 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)
20 simprr 769 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)
21 btwnconn3 35379 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
228, 9, 10, 12, 11, 21syl122anc 1377 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
2322adantr 479 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
2419, 20, 23mp2and 695 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))
2524expr 455 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ (𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
2618, 25jaod 855 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
2726expr 455 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
28 simpll2 1211 . . . . . . . . . . . . . 14 ((((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩) β†’ 𝐡 β‰  𝑃)
2928adantl 480 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐡 β‰  𝑃)
3029necomd 2994 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝑃 β‰  𝐡)
31 simprlr 776 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)
32 simprr 769 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)
33 btwnconn1 35377 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ ((𝑃 β‰  𝐡 ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
348, 9, 11, 10, 12, 33syl122anc 1377 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝑃 β‰  𝐡 ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
3534adantr 479 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ ((𝑃 β‰  𝐡 ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
3630, 31, 32, 35mp3and 1462 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))
3736expr 455 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
38 df-3an 1087 . . . . . . . . . . . 12 (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) ↔ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))
39 simpr3 1194 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)
40 simpr2 1193 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)
418, 9, 12, 11, 10, 39, 40btwnexchand 35302 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)
4241olcd 870 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))
4338, 42sylan2br 593 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))
4443expr 455 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ (𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
4537, 44jaod 855 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
4645expr 455 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) β†’ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ β†’ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
4727, 46jaod 855 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) β†’ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
4847imp32 417 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))
495, 6, 483jca 1126 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝐴 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
5049exp31 418 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) β†’ (((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ (𝐴 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))))
514, 50syl5 34 . . 3 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) β†’ (((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ (𝐴 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩)))))
5251impd 409 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝐴 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
53 broutsideof2 35398 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
548, 9, 10, 11, 53syl13anc 1370 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
55 broutsideof2 35398 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐡, 𝐢⟩ ↔ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))))
568, 9, 11, 12, 55syl13anc 1370 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐡, 𝐢⟩ ↔ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))))
5754, 56anbi12d 629 . . 3 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ 𝑃OutsideOf⟨𝐡, 𝐢⟩) ↔ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)))))
58 df-3an 1087 . . . . 5 ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ↔ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
59 df-3an 1087 . . . . 5 ((𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)) ↔ ((𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)))
6058, 59anbi12i 625 . . . 4 (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))) ↔ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ∧ ((𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))))
61 an4 652 . . . 4 ((((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))) ↔ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ∧ ((𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))))
6260, 61bitr4i 277 . . 3 (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))) ↔ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩))))
6357, 62bitrdi 286 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ 𝑃OutsideOf⟨𝐡, 𝐢⟩) ↔ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐡 β‰  𝑃 ∧ 𝐢 β‰  𝑃)) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐡⟩)))))
64 broutsideof2 35398 . . 3 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐢⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
658, 9, 10, 12, 64syl13anc 1370 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐢⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐢 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐢⟩ ∨ 𝐢 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
6652, 63, 653imtr4d 293 1 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ 𝑃OutsideOf⟨𝐡, 𝐢⟩) β†’ 𝑃OutsideOf⟨𝐴, 𝐢⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   ∈ wcel 2104   β‰  wne 2938  βŸ¨cop 4633   class class class wbr 5147  β€˜cfv 6542  β„•cn 12216  π”Όcee 28413   Btwn cbtwn 28414  OutsideOfcoutsideof 35395
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-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-ee 28416  df-btwn 28417  df-cgr 28418  df-ofs 35259  df-colinear 35315  df-ifs 35316  df-cgr3 35317  df-fs 35318  df-outsideof 35396
This theorem is referenced by: (None)
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