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Theorem pmltpc 24830
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 275 . . . . . 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 5109 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
9 fveq2 6843 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
109breq2d 5118 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝐹𝑤) ≤ (𝐹𝑧) ↔ (𝐹𝑤) ≤ (𝐹𝑥)))
118, 10imbi12d 345 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ (𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥))))
12 breq2 5110 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
13 fveq2 6843 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1413breq1d 5116 . . . . . . . . . 10 (𝑤 = 𝑦 → ((𝐹𝑤) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ (𝐹𝑥)))
1512, 14imbi12d 345 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥)) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
1611, 15cbvral2vw 3226 . . . . . . . 8 (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
177, 16xchbinx 334 . . . . . . 7 (∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
186, 17bitri 275 . . . . . 6 (∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
194, 18anbi12i 628 . . . . 5 ((∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
20 reeanv 3216 . . . . 5 (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))))
21 ioran 983 . . . . 5 (¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))) ↔ (¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
2219, 20, 213bitr4i 303 . . . 4 (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ ¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
23 reeanv 3216 . . . . . 6 (∃𝑦𝐴𝑤𝐴 ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))))
24 simplll 774 . . . . . . . . . 10 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹))
2524simpld 496 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝐹 ∈ (ℝ ↑pm ℝ))
2624simprd 497 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝐴 ⊆ dom 𝐹)
27 simpllr 775 . . . . . . . . . 10 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → (𝑥𝐴𝑧𝐴))
2827simpld 496 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑥𝐴)
29 simplrl 776 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑦𝐴)
3027simprd 497 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑧𝐴)
31 simplrr 777 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑤𝐴)
32 simprll 778 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑥𝑦)
33 simprrl 780 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑧𝑤)
34 simprlr 779 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → ¬ (𝐹𝑥) ≤ (𝐹𝑦))
35 simprrr 781 . . . . . . . . 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 24829 . . . . . . . 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 414 . . . . . . 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 3202 . . . . . 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 3202 . . . 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 862 . 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 1089 . 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 397  wo 846  w3o 1087  w3a 1088  wcel 2107  wral 3061  wrex 3070  wss 3911   class class class wbr 5106  dom cdm 5634  cfv 6497  (class class class)co 7358  pm cpm 8769  cr 11055   < clt 11194  cle 11195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-pre-lttri 11130  ax-pre-lttrn 11131
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8651  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200
This theorem is referenced by: (None)
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