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Theorem pmltpc 23419
 Description: Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)
Assertion
Ref Expression
pmltpc ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)) ∨ ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑥,𝑦,𝐴   𝐹,𝑎,𝑏,𝑐,𝑥,𝑦

Proof of Theorem pmltpc
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexanali 3136 . . . . . . . 8 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
21rexbii 3179 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
3 rexnal 3133 . . . . . . 7 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
42, 3bitri 264 . . . . . 6 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
5 rexanali 3136 . . . . . . . 8 (∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
65rexbii 3179 . . . . . . 7 (∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
7 rexnal 3133 . . . . . . . 8 (∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
8 breq1 4807 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
9 fveq2 6352 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
109breq2d 4816 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝐹𝑤) ≤ (𝐹𝑧) ↔ (𝐹𝑤) ≤ (𝐹𝑥)))
118, 10imbi12d 333 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ (𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥))))
12 breq2 4808 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
13 fveq2 6352 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1413breq1d 4814 . . . . . . . . . 10 (𝑤 = 𝑦 → ((𝐹𝑤) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ (𝐹𝑥)))
1512, 14imbi12d 333 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥)) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
1611, 15cbvral2v 3318 . . . . . . . 8 (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
177, 16xchbinx 323 . . . . . . 7 (∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
186, 17bitri 264 . . . . . 6 (∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
194, 18anbi12i 735 . . . . 5 ((∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
20 reeanv 3245 . . . . 5 (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))))
21 ioran 512 . . . . 5 (¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))) ↔ (¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
2219, 20, 213bitr4i 292 . . . 4 (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ ¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
23 reeanv 3245 . . . . . 6 (∃𝑦𝐴𝑤𝐴 ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))))
24 simplll 815 . . . . . . . . . 10 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹))
2524simpld 477 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝐹 ∈ (ℝ ↑pm ℝ))
2624simprd 482 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝐴 ⊆ dom 𝐹)
27 simpllr 817 . . . . . . . . . 10 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → (𝑥𝐴𝑧𝐴))
2827simpld 477 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑥𝐴)
29 simplrl 819 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑦𝐴)
3027simprd 482 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑧𝐴)
31 simplrr 820 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑤𝐴)
32 simprll 821 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑥𝑦)
33 simprrl 823 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑧𝑤)
34 simprlr 822 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → ¬ (𝐹𝑥) ≤ (𝐹𝑦))
35 simprrr 824 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → ¬ (𝐹𝑤) ≤ (𝐹𝑧))
3625, 26, 28, 29, 30, 31, 32, 33, 34, 35pmltpclem2 23418 . . . . . . . 8 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐)))))
3736ex 449 . . . . . . 7 ((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) → (((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
3837rexlimdvva 3176 . . . . . 6 (((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐴𝑤𝐴 ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
3923, 38syl5bir 233 . . . . 5 (((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
4039rexlimdvva 3176 . . . 4 ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
4122, 40syl5bir 233 . . 3 ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))) → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
4241orrd 392 . 2 ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))) ∨ ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
43 df-3or 1073 . 2 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)) ∨ ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))) ↔ ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))) ∨ ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
4442, 43sylibr 224 1 ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)) ∨ ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1071   ∧ w3a 1072   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ⊆ wss 3715   class class class wbr 4804  dom cdm 5266  ‘cfv 6049  (class class class)co 6813   ↑pm cpm 8024  ℝcr 10127   < clt 10266   ≤ cle 10267 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-pre-lttri 10202  ax-pre-lttrn 10203 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-er 7911  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272 This theorem is referenced by: (None)
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