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Theorem pmltpc 24959
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 3103 . . . . . . . 8 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
21rexbii 3095 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
3 rexnal 3101 . . . . . . 7 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
42, 3bitri 275 . . . . . 6 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
5 rexanali 3103 . . . . . . . 8 (∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
65rexbii 3095 . . . . . . 7 (∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
7 rexnal 3101 . . . . . . . 8 (∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
8 breq1 5151 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
9 fveq2 6889 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
109breq2d 5160 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝐹𝑤) ≤ (𝐹𝑧) ↔ (𝐹𝑤) ≤ (𝐹𝑥)))
118, 10imbi12d 345 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ (𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥))))
12 breq2 5152 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
13 fveq2 6889 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1413breq1d 5158 . . . . . . . . . 10 (𝑤 = 𝑦 → ((𝐹𝑤) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ (𝐹𝑥)))
1512, 14imbi12d 345 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥)) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
1611, 15cbvral2vw 3239 . . . . . . . 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 3227 . . . . 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 3227 . . . . . 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 24958 . . . . . . . 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 3212 . . . . . 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 3212 . . . 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 3062  wrex 3071  wss 3948   class class class wbr 5148  dom cdm 5676  cfv 6541  (class class class)co 7406  pm cpm 8818  cr 11106   < clt 11245  cle 11246
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-pre-lttri 11181  ax-pre-lttrn 11182
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-er 8700  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251
This theorem is referenced by: (None)
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