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Axiom ax-ddkcomp 10114
 Description: Axiom of Dedekind completeness for Dedekind real numbers: every nonempty upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then the axiom ax-ddkcomp 10114 should be used in place of construction specific results. In particular, axcaucvg 6974 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
Assertion
Ref Expression
ax-ddkcomp (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 𝑦𝑥 ∧ ((𝐵𝑅 ∧ ∀𝑦𝐴 𝑦𝐵) → 𝑥𝐵)))

Detailed syntax breakdown of Axiom ax-ddkcomp
StepHypRef Expression
1 cA . . . . 5 class 𝐴
2 cr 6888 . . . . 5 class
31, 2wss 2917 . . . 4 wff 𝐴 ⊆ ℝ
4 c0 3224 . . . . 5 class
51, 4wne 2204 . . . 4 wff 𝐴 ≠ ∅
63, 5wa 97 . . 3 wff (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅)
7 vy . . . . . . 7 setvar 𝑦
87cv 1242 . . . . . 6 class 𝑦
9 vx . . . . . . 7 setvar 𝑥
109cv 1242 . . . . . 6 class 𝑥
11 clt 7060 . . . . . 6 class <
128, 10, 11wbr 3764 . . . . 5 wff 𝑦 < 𝑥
1312, 7, 1wral 2306 . . . 4 wff 𝑦𝐴 𝑦 < 𝑥
1413, 9, 2wrex 2307 . . 3 wff 𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥
1510, 8, 11wbr 3764 . . . . . 6 wff 𝑥 < 𝑦
16 vz . . . . . . . . . 10 setvar 𝑧
1716cv 1242 . . . . . . . . 9 class 𝑧
1810, 17, 11wbr 3764 . . . . . . . 8 wff 𝑥 < 𝑧
1918, 16, 1wrex 2307 . . . . . . 7 wff 𝑧𝐴 𝑥 < 𝑧
2017, 8, 11wbr 3764 . . . . . . . 8 wff 𝑧 < 𝑦
2120, 16, 1wral 2306 . . . . . . 7 wff 𝑧𝐴 𝑧 < 𝑦
2219, 21wo 629 . . . . . 6 wff (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)
2315, 22wi 4 . . . . 5 wff (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))
2423, 7, 2wral 2306 . . . 4 wff 𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))
2524, 9, 2wral 2306 . . 3 wff 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))
266, 14, 25w3a 885 . 2 wff ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))
27 cle 7061 . . . . . 6 class
288, 10, 27wbr 3764 . . . . 5 wff 𝑦𝑥
2928, 7, 1wral 2306 . . . 4 wff 𝑦𝐴 𝑦𝑥
30 cB . . . . . . 7 class 𝐵
31 cR . . . . . . 7 class 𝑅
3230, 31wcel 1393 . . . . . 6 wff 𝐵𝑅
338, 30, 27wbr 3764 . . . . . . 7 wff 𝑦𝐵
3433, 7, 1wral 2306 . . . . . 6 wff 𝑦𝐴 𝑦𝐵
3532, 34wa 97 . . . . 5 wff (𝐵𝑅 ∧ ∀𝑦𝐴 𝑦𝐵)
3610, 30, 27wbr 3764 . . . . 5 wff 𝑥𝐵
3735, 36wi 4 . . . 4 wff ((𝐵𝑅 ∧ ∀𝑦𝐴 𝑦𝐵) → 𝑥𝐵)
3829, 37wa 97 . . 3 wff (∀𝑦𝐴 𝑦𝑥 ∧ ((𝐵𝑅 ∧ ∀𝑦𝐴 𝑦𝐵) → 𝑥𝐵))
3938, 9, 2wrex 2307 . 2 wff 𝑥 ∈ ℝ (∀𝑦𝐴 𝑦𝑥 ∧ ((𝐵𝑅 ∧ ∀𝑦𝐴 𝑦𝐵) → 𝑥𝐵))
4026, 39wi 4 1 wff (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 𝑦𝑥 ∧ ((𝐵𝑅 ∧ ∀𝑦𝐴 𝑦𝐵) → 𝑥𝐵)))
 Colors of variables: wff set class This axiom is referenced by: (None)
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