Theorem axpre-suploc 7710

Description: An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given 𝑥 < 𝑦, either there is an element of the set greater than 𝑥, or 𝑦 is an upper bound.

This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7741. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-suploc	⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))

Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem axpre-suploc

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 519	. . 3 ⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 𝑥 ∈ 𝐴)
2		eleq1w 2200	. . . 4 ⊢ (𝑥 = 𝑑 → (𝑥 ∈ 𝐴 ↔ 𝑑 ∈ 𝐴))
3	2	cbvexv 1890	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑑 𝑑 ∈ 𝐴)
4	1, 3	sylib 121	. 2 ⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑑 𝑑 ∈ 𝐴)
5		simplll 522	. . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) ∧ 𝑑 ∈ 𝐴) → 𝐴 ⊆ ℝ)
6		simpr 109	. . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ 𝐴)
7		simplrl 524	. . . . 5 ⊢ ((((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) ∧ 𝑑 ∈ 𝐴) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
8		breq2 3933	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑏 <ℝ 𝑎 ↔ 𝑏 <ℝ 𝑥))
9	8	ralbidv 2437	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥))
10	9	cbvrexv 2655	. . . . . 6 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥)
11		breq1 3932	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑏 <ℝ 𝑥 ↔ 𝑦 <ℝ 𝑥))
12	11	cbvralv 2654	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
13	12	rexbii 2442	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
14	10, 13	bitri 183	. . . . 5 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
15	7, 14	sylibr 133	. . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) ∧ 𝑑 ∈ 𝐴) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎)
16		simplrr 525	. . . . 5 ⊢ ((((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) ∧ 𝑑 ∈ 𝐴) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
17		breq1 3932	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑎 <ℝ 𝑏 ↔ 𝑥 <ℝ 𝑏))
18		breq1 3932	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 <ℝ 𝑐 ↔ 𝑥 <ℝ 𝑐))
19	18	rexbidv 2438	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ↔ ∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐))
20	19	orbi1d 780	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ((∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏) ↔ (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏)))
21	17, 20	imbi12d 233	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝑎 <ℝ 𝑏 → (∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏)) ↔ (𝑥 <ℝ 𝑏 → (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏))))
22		breq2 3933	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑥 <ℝ 𝑏 ↔ 𝑥 <ℝ 𝑦))
23		breq2 3933	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑐 <ℝ 𝑏 ↔ 𝑐 <ℝ 𝑦))
24	23	ralbidv 2437	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏 ↔ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦))
25	24	orbi2d 779	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏) ↔ (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦)))
26	22, 25	imbi12d 233	. . . . . . 7 ⊢ (𝑏 = 𝑦 → ((𝑥 <ℝ 𝑏 → (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏)) ↔ (𝑥 <ℝ 𝑦 → (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦))))
27	21, 26	cbvral2v 2665	. . . . . 6 ⊢ (∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ (𝑎 <ℝ 𝑏 → (∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏)) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦)))
28		breq2 3933	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑥 <ℝ 𝑐 ↔ 𝑥 <ℝ 𝑧))
29	28	cbvrexv 2655	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ↔ ∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧)
30		breq1 3932	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑐 <ℝ 𝑦 ↔ 𝑧 <ℝ 𝑦))
31	30	cbvralv 2654	. . . . . . . . 9 ⊢ (∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)
32	29, 31	orbi12i 753	. . . . . . . 8 ⊢ ((∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦) ↔ (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦))
33	32	imbi2i 225	. . . . . . 7 ⊢ ((𝑥 <ℝ 𝑦 → (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦)) ↔ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
34	33	2ralbii 2443	. . . . . 6 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦)) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
35	27, 34	bitri 183	. . . . 5 ⊢ (∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ (𝑎 <ℝ 𝑏 → (∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏)) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
36	16, 35	sylibr 133	. . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) ∧ 𝑑 ∈ 𝐴) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ (𝑎 <ℝ 𝑏 → (∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏)))
37		eqid 2139	. . . 4 ⊢ {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴} = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}
38	5, 6, 15, 36, 37	axpre-suploclemres 7709	. . 3 ⊢ ((((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) ∧ 𝑑 ∈ 𝐴) → ∃𝑎 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑎 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)))
39	17	notbid 656	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (¬ 𝑎 <ℝ 𝑏 ↔ ¬ 𝑥 <ℝ 𝑏))
40	39	ralbidv 2437	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝐴 ¬ 𝑎 <ℝ 𝑏 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏))
41	8	imbi1d 230	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ((𝑏 <ℝ 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐) ↔ (𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)))
42	41	ralbidv 2437	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐) ↔ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)))
43	40, 42	anbi12d 464	. . . . . 6 ⊢ (𝑎 = 𝑥 → ((∀𝑏 ∈ 𝐴 ¬ 𝑎 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐))))
44	43	cbvrexv 2655	. . . . 5 ⊢ (∃𝑎 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑎 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)) ↔ ∃𝑥 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)))
45	22	notbid 656	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (¬ 𝑥 <ℝ 𝑏 ↔ ¬ 𝑥 <ℝ 𝑦))
46	45	cbvralv 2654	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦)
47		breq1 3932	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑏 <ℝ 𝑐 ↔ 𝑦 <ℝ 𝑐))
48	47	rexbidv 2438	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐 ↔ ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐))
49	11, 48	imbi12d 233	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐) ↔ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐)))
50	49	cbvralv 2654	. . . . . . 7 ⊢ (∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐) ↔ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐))
51	46, 50	anbi12i 455	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐)))
52	51	rexbii 2442	. . . . 5 ⊢ (∃𝑥 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)) ↔ ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐)))
53	44, 52	bitri 183	. . . 4 ⊢ (∃𝑎 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑎 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)) ↔ ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐)))
54		breq2 3933	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝑦 <ℝ 𝑐 ↔ 𝑦 <ℝ 𝑧))
55	54	cbvrexv 2655	. . . . . . . 8 ⊢ (∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐 ↔ ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)
56	55	imbi2i 225	. . . . . . 7 ⊢ ((𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐) ↔ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
57	56	ralbii 2441	. . . . . 6 ⊢ (∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐) ↔ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
58	57	anbi2i 452	. . . . 5 ⊢ ((∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
59	58	rexbii 2442	. . . 4 ⊢ (∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐)) ↔ ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
60	53, 59	bitri 183	. . 3 ⊢ (∃𝑎 ∈ ℝ (∀𝑏 ∈ 𝐴 ¬ 𝑎 <ℝ 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 <ℝ 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)) ↔ ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
61	38, 60	sylib 121	. 2 ⊢ ((((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) ∧ 𝑑 ∈ 𝐴) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
62	4, 61	exlimddv 1870	1 ⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929  Rcnr 7105  0_Rc0r 7106  ℝcr 7619   <_ℝ cltrr 7624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-imp 7277  df-iltp 7278  df-enr 7534  df-nr 7535  df-plr 7536  df-mr 7537  df-ltr 7538  df-0r 7539  df-1r 7540  df-m1r 7541  df-r 7630  df-lt 7633

This theorem is referenced by: (None)