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Theorem pmltpc 24966
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 3102 . . . . . . . 8 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
21rexbii 3094 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
3 rexnal 3100 . . . . . . 7 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
42, 3bitri 274 . . . . . 6 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
5 rexanali 3102 . . . . . . . 8 (∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
65rexbii 3094 . . . . . . 7 (∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
7 rexnal 3100 . . . . . . . 8 (∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
8 breq1 5151 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
9 fveq2 6891 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
109breq2d 5160 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝐹𝑤) ≤ (𝐹𝑧) ↔ (𝐹𝑤) ≤ (𝐹𝑥)))
118, 10imbi12d 344 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ (𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥))))
12 breq2 5152 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
13 fveq2 6891 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1413breq1d 5158 . . . . . . . . . 10 (𝑤 = 𝑦 → ((𝐹𝑤) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ (𝐹𝑥)))
1512, 14imbi12d 344 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥)) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
1611, 15cbvral2vw 3238 . . . . . . . 8 (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
177, 16xchbinx 333 . . . . . . 7 (∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
186, 17bitri 274 . . . . . 6 (∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
194, 18anbi12i 627 . . . . 5 ((∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
20 reeanv 3226 . . . . 5 (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))))
21 ioran 982 . . . . 5 (¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))) ↔ (¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
2219, 20, 213bitr4i 302 . . . 4 (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ ¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
23 reeanv 3226 . . . . . 6 (∃𝑦𝐴𝑤𝐴 ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))))
24 simplll 773 . . . . . . . . . 10 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹))
2524simpld 495 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝐹 ∈ (ℝ ↑pm ℝ))
2624simprd 496 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝐴 ⊆ dom 𝐹)
27 simpllr 774 . . . . . . . . . 10 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → (𝑥𝐴𝑧𝐴))
2827simpld 495 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑥𝐴)
29 simplrl 775 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑦𝐴)
3027simprd 496 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑧𝐴)
31 simplrr 776 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑤𝐴)
32 simprll 777 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑥𝑦)
33 simprrl 779 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑧𝑤)
34 simprlr 778 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → ¬ (𝐹𝑥) ≤ (𝐹𝑦))
35 simprrr 780 . . . . . . . . 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 24965 . . . . . . . 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 413 . . . . . . 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 3211 . . . . . 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, 38biimtrrid 242 . . . . 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 3211 . . . 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, 40biimtrrid 242 . . 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 861 . 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 1088 . 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 233 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  wa 396  wo 845  w3o 1086  w3a 1087  wcel 2106  wral 3061  wrex 3070  wss 3948   class class class wbr 5148  dom cdm 5676  cfv 6543  (class class class)co 7408  pm cpm 8820  cr 11108   < clt 11247  cle 11248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-pre-lttri 11183  ax-pre-lttrn 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-er 8702  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253
This theorem is referenced by: (None)
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