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Theorem pmltpc 23754
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 3211 . . . . . . . 8 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
21rexbii 3194 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
3 rexnal 3185 . . . . . . 7 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
42, 3bitri 267 . . . . . 6 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
5 rexanali 3211 . . . . . . . 8 (∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
65rexbii 3194 . . . . . . 7 (∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
7 rexnal 3185 . . . . . . . 8 (∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)))
8 breq1 4932 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
9 fveq2 6499 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
109breq2d 4941 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝐹𝑤) ≤ (𝐹𝑧) ↔ (𝐹𝑤) ≤ (𝐹𝑥)))
118, 10imbi12d 337 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ (𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥))))
12 breq2 4933 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
13 fveq2 6499 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1413breq1d 4939 . . . . . . . . . 10 (𝑤 = 𝑦 → ((𝐹𝑤) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ (𝐹𝑥)))
1512, 14imbi12d 337 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑥𝑤 → (𝐹𝑤) ≤ (𝐹𝑥)) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
1611, 15cbvral2v 3392 . . . . . . . 8 (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
177, 16xchbinx 326 . . . . . . 7 (∃𝑧𝐴 ¬ ∀𝑤𝐴 (𝑧𝑤 → (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
186, 17bitri 267 . . . . . 6 (∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))
194, 18anbi12i 617 . . . . 5 ((∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
20 reeanv 3308 . . . . 5 (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑧𝐴𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))))
21 ioran 966 . . . . 5 (¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))) ↔ (¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ¬ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
2219, 20, 213bitr4i 295 . . . 4 (∃𝑥𝐴𝑧𝐴 (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ ¬ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥))))
23 reeanv 3308 . . . . . 6 (∃𝑦𝐴𝑤𝐴 ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))) ↔ (∃𝑦𝐴 (𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ ∃𝑤𝐴 (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧))))
24 simplll 762 . . . . . . . . . 10 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹))
2524simpld 487 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝐹 ∈ (ℝ ↑pm ℝ))
2624simprd 488 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝐴 ⊆ dom 𝐹)
27 simpllr 763 . . . . . . . . . 10 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → (𝑥𝐴𝑧𝐴))
2827simpld 487 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑥𝐴)
29 simplrl 764 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑦𝐴)
3027simprd 488 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑧𝐴)
31 simplrr 765 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑤𝐴)
32 simprll 766 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑥𝑦)
33 simprrl 768 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → 𝑧𝑤)
34 simprlr 767 . . . . . . . . 9 (((((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐴𝑤𝐴)) ∧ ((𝑥𝑦 ∧ ¬ (𝐹𝑥) ≤ (𝐹𝑦)) ∧ (𝑧𝑤 ∧ ¬ (𝐹𝑤) ≤ (𝐹𝑧)))) → ¬ (𝐹𝑥) ≤ (𝐹𝑦))
35 simprrr 769 . . . . . . . . 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 23753 . . . . . . . 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 405 . . . . . . 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 3239 . . . . . 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 235 . . . . 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 3239 . . . 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 235 . . 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 849 . 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 1069 . 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 226 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 387  wo 833  w3o 1067  w3a 1068  wcel 2050  wral 3088  wrex 3089  wss 3829   class class class wbr 4929  dom cdm 5407  cfv 6188  (class class class)co 6976  pm cpm 8207  cr 10334   < clt 10474  cle 10475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-pre-lttri 10409  ax-pre-lttrn 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-po 5326  df-so 5327  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-er 8089  df-pm 8209  df-en 8307  df-dom 8308  df-sdom 8309  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480
This theorem is referenced by: (None)
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