Theorem axpre-suploc 7969

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 8000. (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 528	. . 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 2257	. . . 4 ⊢ (𝑥 = 𝑑 → (𝑥 ∈ 𝐴 ↔ 𝑑 ∈ 𝐴))
3	2	cbvexv 1933	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑑 𝑑 ∈ 𝐴)
4	1, 3	sylib 122	. 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 533	. . . 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 110	. . . 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 535	. . . . 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 4037	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑏 <ℝ 𝑎 ↔ 𝑏 <ℝ 𝑥))
9	8	ralbidv 2497	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥))
10	9	cbvrexv 2730	. . . . . 6 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥)
11		breq1 4036	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑏 <ℝ 𝑥 ↔ 𝑦 <ℝ 𝑥))
12	11	cbvralv 2729	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
13	12	rexbii 2504	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
14	10, 13	bitri 184	. . . . 5 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
15	7, 14	sylibr 134	. . . 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 536	. . . . 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 4036	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑎 <ℝ 𝑏 ↔ 𝑥 <ℝ 𝑏))
18		breq1 4036	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 <ℝ 𝑐 ↔ 𝑥 <ℝ 𝑐))
19	18	rexbidv 2498	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ↔ ∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐))
20	19	orbi1d 792	. . . . . . . 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 234	. . . . . . 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 4037	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑥 <ℝ 𝑏 ↔ 𝑥 <ℝ 𝑦))
23		breq2 4037	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑐 <ℝ 𝑏 ↔ 𝑐 <ℝ 𝑦))
24	23	ralbidv 2497	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏 ↔ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦))
25	24	orbi2d 791	. . . . . . . 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 234	. . . . . . 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 2742	. . . . . 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 4037	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑥 <ℝ 𝑐 ↔ 𝑥 <ℝ 𝑧))
29	28	cbvrexv 2730	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ↔ ∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧)
30		breq1 4036	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑐 <ℝ 𝑦 ↔ 𝑧 <ℝ 𝑦))
31	30	cbvralv 2729	. . . . . . . . 9 ⊢ (∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)
32	29, 31	orbi12i 765	. . . . . . . 8 ⊢ ((∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ∨ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦) ↔ (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦))
33	32	imbi2i 226	. . . . . . 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 2505	. . . . . 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 184	. . . . 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 134	. . . 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 2196	. . . 4 ⊢ {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴} = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}
38	5, 6, 15, 36, 37	axpre-suploclemres 7968	. . 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 668	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (¬ 𝑎 <ℝ 𝑏 ↔ ¬ 𝑥 <ℝ 𝑏))
40	39	ralbidv 2497	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝐴 ¬ 𝑎 <ℝ 𝑏 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏))
41	8	imbi1d 231	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ((𝑏 <ℝ 𝑎 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐) ↔ (𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐)))
42	41	ralbidv 2497	. . . . . . 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 473	. . . . . 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 2730	. . . . 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 668	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (¬ 𝑥 <ℝ 𝑏 ↔ ¬ 𝑥 <ℝ 𝑦))
46	45	cbvralv 2729	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦)
47		breq1 4036	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑏 <ℝ 𝑐 ↔ 𝑦 <ℝ 𝑐))
48	47	rexbidv 2498	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐 ↔ ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐))
49	11, 48	imbi12d 234	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝑏 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑏 <ℝ 𝑐) ↔ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐)))
50	49	cbvralv 2729	. . . . . . 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 460	. . . . . 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 2504	. . . . 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 184	. . . 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 4037	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝑦 <ℝ 𝑐 ↔ 𝑦 <ℝ 𝑧))
55	54	cbvrexv 2730	. . . . . . . 8 ⊢ (∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐 ↔ ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)
56	55	imbi2i 226	. . . . . . 7 ⊢ ((𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐) ↔ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
57	56	ralbii 2503	. . . . . 6 ⊢ (∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑐 ∈ 𝐴 𝑦 <ℝ 𝑐) ↔ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
58	57	anbi2i 457	. . . . 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 2504	. . . 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 184	. . 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 122	. 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 1913	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 104   ∨ wo 709  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479   ⊆ wss 3157  ⟨cop 3625   class class class wbr 4033  Rcnr 7364  0_Rc0r 7365  ℝcr 7878   <_ℝ cltrr 7883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-imp 7536  df-iltp 7537  df-enr 7793  df-nr 7794  df-plr 7795  df-mr 7796  df-ltr 7797  df-0r 7798  df-1r 7799  df-m1r 7800  df-r 7889  df-lt 7892

This theorem is referenced by: (None)