Theorem axpre-suploc 8077

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 8108. (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 2290	. . . 4 ⊢ (𝑥 = 𝑑 → (𝑥 ∈ 𝐴 ↔ 𝑑 ∈ 𝐴))
3	2	cbvexv 1965	. . 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 4086	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑏 <ℝ 𝑎 ↔ 𝑏 <ℝ 𝑥))
9	8	ralbidv 2530	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥))
10	9	cbvrexv 2766	. . . . . 6 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥)
11		breq1 4085	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑏 <ℝ 𝑥 ↔ 𝑦 <ℝ 𝑥))
12	11	cbvralv 2765	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
13	12	rexbii 2537	. . . . . 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 4085	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑎 <ℝ 𝑏 ↔ 𝑥 <ℝ 𝑏))
18		breq1 4085	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 <ℝ 𝑐 ↔ 𝑥 <ℝ 𝑐))
19	18	rexbidv 2531	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ↔ ∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐))
20	19	orbi1d 796	. . . . . . . 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 4086	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑥 <ℝ 𝑏 ↔ 𝑥 <ℝ 𝑦))
23		breq2 4086	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑐 <ℝ 𝑏 ↔ 𝑐 <ℝ 𝑦))
24	23	ralbidv 2530	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏 ↔ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦))
25	24	orbi2d 795	. . . . . . . 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 2778	. . . . . 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 4086	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑥 <ℝ 𝑐 ↔ 𝑥 <ℝ 𝑧))
29	28	cbvrexv 2766	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ↔ ∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧)
30		breq1 4085	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑐 <ℝ 𝑦 ↔ 𝑧 <ℝ 𝑦))
31	30	cbvralv 2765	. . . . . . . . 9 ⊢ (∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)
32	29, 31	orbi12i 769	. . . . . . . 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 2538	. . . . . 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 2229	. . . 4 ⊢ {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴} = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}
38	5, 6, 15, 36, 37	axpre-suploclemres 8076	. . 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 671	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (¬ 𝑎 <ℝ 𝑏 ↔ ¬ 𝑥 <ℝ 𝑏))
40	39	ralbidv 2530	. . . . . . 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 2530	. . . . . . 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 2766	. . . . 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 671	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (¬ 𝑥 <ℝ 𝑏 ↔ ¬ 𝑥 <ℝ 𝑦))
46	45	cbvralv 2765	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦)
47		breq1 4085	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑏 <ℝ 𝑐 ↔ 𝑦 <ℝ 𝑐))
48	47	rexbidv 2531	. . . . . . . . 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 2765	. . . . . . 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 2537	. . . . 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 4086	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝑦 <ℝ 𝑐 ↔ 𝑦 <ℝ 𝑧))
55	54	cbvrexv 2766	. . . . . . . 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 2536	. . . . . 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 2537	. . . 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 1945	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 713  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {crab 2512   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4082  Rcnr 7472  0_Rc0r 7473  ℝcr 7986   <_ℝ cltrr 7991

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4377  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-1o 6552  df-2o 6553  df-oadd 6556  df-omul 6557  df-er 6670  df-ec 6672  df-qs 6676  df-ni 7479  df-pli 7480  df-mi 7481  df-lti 7482  df-plpq 7519  df-mpq 7520  df-enq 7522  df-nqqs 7523  df-plqqs 7524  df-mqqs 7525  df-1nqqs 7526  df-rq 7527  df-ltnqqs 7528  df-enq0 7599  df-nq0 7600  df-0nq0 7601  df-plq0 7602  df-mq0 7603  df-inp 7641  df-i1p 7642  df-iplp 7643  df-imp 7644  df-iltp 7645  df-enr 7901  df-nr 7902  df-plr 7903  df-mr 7904  df-ltr 7905  df-0r 7906  df-1r 7907  df-m1r 7908  df-r 7997  df-lt 8000

This theorem is referenced by: (None)