Theorem axpre-suploc 8122

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 8153. (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 529	. . 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 2292	. . . 4 ⊢ (𝑥 = 𝑑 → (𝑥 ∈ 𝐴 ↔ 𝑑 ∈ 𝐴))
3	2	cbvexv 1967	. . 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 535	. . . 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 537	. . . . 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 4092	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑏 <ℝ 𝑎 ↔ 𝑏 <ℝ 𝑥))
9	8	ralbidv 2532	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥))
10	9	cbvrexv 2768	. . . . . 6 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥)
11		breq1 4091	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑏 <ℝ 𝑥 ↔ 𝑦 <ℝ 𝑥))
12	11	cbvralv 2767	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝐴 𝑏 <ℝ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
13	12	rexbii 2539	. . . . . 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 538	. . . . 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 4091	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑎 <ℝ 𝑏 ↔ 𝑥 <ℝ 𝑏))
18		breq1 4091	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 <ℝ 𝑐 ↔ 𝑥 <ℝ 𝑐))
19	18	rexbidv 2533	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃𝑐 ∈ 𝐴 𝑎 <ℝ 𝑐 ↔ ∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐))
20	19	orbi1d 798	. . . . . . . 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 4092	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (𝑥 <ℝ 𝑏 ↔ 𝑥 <ℝ 𝑦))
23		breq2 4092	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑐 <ℝ 𝑏 ↔ 𝑐 <ℝ 𝑦))
24	23	ralbidv 2532	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑏 ↔ ∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦))
25	24	orbi2d 797	. . . . . . . 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 2780	. . . . . 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 4092	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑥 <ℝ 𝑐 ↔ 𝑥 <ℝ 𝑧))
29	28	cbvrexv 2768	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝐴 𝑥 <ℝ 𝑐 ↔ ∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧)
30		breq1 4091	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑐 <ℝ 𝑦 ↔ 𝑧 <ℝ 𝑦))
31	30	cbvralv 2767	. . . . . . . . 9 ⊢ (∀𝑐 ∈ 𝐴 𝑐 <ℝ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)
32	29, 31	orbi12i 771	. . . . . . . 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 2540	. . . . . 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 2231	. . . 4 ⊢ {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴} = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}
38	5, 6, 15, 36, 37	axpre-suploclemres 8121	. . 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 673	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (¬ 𝑎 <ℝ 𝑏 ↔ ¬ 𝑥 <ℝ 𝑏))
40	39	ralbidv 2532	. . . . . . 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 2532	. . . . . . 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 2768	. . . . 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 673	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → (¬ 𝑥 <ℝ 𝑏 ↔ ¬ 𝑥 <ℝ 𝑦))
46	45	cbvralv 2767	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑏 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦)
47		breq1 4091	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑏 <ℝ 𝑐 ↔ 𝑦 <ℝ 𝑐))
48	47	rexbidv 2533	. . . . . . . . 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 2767	. . . . . . 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 2539	. . . . 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 4092	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝑦 <ℝ 𝑐 ↔ 𝑦 <ℝ 𝑧))
55	54	cbvrexv 2768	. . . . . . . 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 2538	. . . . . 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 2539	. . . 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 1947	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 715  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088  Rcnr 7517  0_Rc0r 7518  ℝcr 8031   <_ℝ cltrr 8036

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-i1p 7687  df-iplp 7688  df-imp 7689  df-iltp 7690  df-enr 7946  df-nr 7947  df-plr 7948  df-mr 7949  df-ltr 7950  df-0r 7951  df-1r 7952  df-m1r 7953  df-r 8042  df-lt 8045

This theorem is referenced by: (None)