Theorem axpre-suploclemres 7733

Description: Lemma for axpre-suploc 7734. The result. The proof just needs to define 𝐵 as basically the same set as 𝐴 (but expressed as a subset of R rather than a subset of ℝ), and apply suplocsr 7641. (Contributed by Jim Kingdon, 24-Jan-2024.)

Hypotheses

Ref	Expression
axpre-suploclem.ss	⊢ (𝜑 → 𝐴 ⊆ ℝ)
axpre-suploclem.m	⊢ (𝜑 → 𝐶 ∈ 𝐴)
axpre-suploclem.ub	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
axpre-suploclem.loc	⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
axpre-suploclem.b	⊢ 𝐵 = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}

Assertion

Ref	Expression
axpre-suploclemres	⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))

Distinct variable groups:   𝑦,𝐴,𝑧,𝑤   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧,𝑥   𝑤,𝐶   𝜑,𝑦,𝑧,𝑥

Allowed substitution hints:   𝜑(𝑤)   𝐵(𝑤)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem axpre-suploclemres

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

Step	Hyp	Ref	Expression
1		axpre-suploclem.ss	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		axpre-suploclem.m	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3	1, 2	sseldd 3103	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4		elreal2 7662	. . . . . . 7 ⊢ (𝐶 ∈ ℝ ↔ ((1st ‘𝐶) ∈ R ∧ 𝐶 = ⟨(1st ‘𝐶), 0R⟩))
5	3, 4	sylib 121	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐶) ∈ R ∧ 𝐶 = ⟨(1st ‘𝐶), 0R⟩))
6	5	simpld 111	. . . . 5 ⊢ (𝜑 → (1st ‘𝐶) ∈ R)
7	5	simprd 113	. . . . . 6 ⊢ (𝜑 → 𝐶 = ⟨(1st ‘𝐶), 0R⟩)
8	7, 2	eqeltrrd 2218	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘𝐶), 0R⟩ ∈ 𝐴)
9		opeq1 3713	. . . . . . 7 ⊢ (𝑤 = (1st ‘𝐶) → ⟨𝑤, 0R⟩ = ⟨(1st ‘𝐶), 0R⟩)
10	9	eleq1d 2209	. . . . . 6 ⊢ (𝑤 = (1st ‘𝐶) → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨(1st ‘𝐶), 0R⟩ ∈ 𝐴))
11		axpre-suploclem.b	. . . . . 6 ⊢ 𝐵 = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}
12	10, 11	elrab2 2847	. . . . 5 ⊢ ((1st ‘𝐶) ∈ 𝐵 ↔ ((1st ‘𝐶) ∈ R ∧ ⟨(1st ‘𝐶), 0R⟩ ∈ 𝐴))
13	6, 8, 12	sylanbrc 414	. . . 4 ⊢ (𝜑 → (1st ‘𝐶) ∈ 𝐵)
14		eleq1 2203	. . . . 5 ⊢ (𝑎 = (1st ‘𝐶) → (𝑎 ∈ 𝐵 ↔ (1st ‘𝐶) ∈ 𝐵))
15	14	spcegv 2777	. . . 4 ⊢ ((1st ‘𝐶) ∈ 𝐵 → ((1st ‘𝐶) ∈ 𝐵 → ∃𝑎 𝑎 ∈ 𝐵))
16	13, 13, 15	sylc 62	. . 3 ⊢ (𝜑 → ∃𝑎 𝑎 ∈ 𝐵)
17		axpre-suploclem.ub	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
18		simprl 521	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → 𝑥 ∈ ℝ)
19		elreal2 7662	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↔ ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
20	18, 19	sylib 121	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
21	20	simpld 111	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (1st ‘𝑥) ∈ R)
22		breq1 3940	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → (𝑦 <ℝ 𝑥 ↔ ⟨𝑏, 0R⟩ <ℝ 𝑥))
23		simplrr 526	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
24		simpr 109	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
25		opeq1 3713	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑏 → ⟨𝑤, 0R⟩ = ⟨𝑏, 0R⟩)
26	25	eleq1d 2209	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑏 → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨𝑏, 0R⟩ ∈ 𝐴))
27	26, 11	elrab2 2847	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ R ∧ ⟨𝑏, 0R⟩ ∈ 𝐴))
28	24, 27	sylib 121	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → (𝑏 ∈ R ∧ ⟨𝑏, 0R⟩ ∈ 𝐴))
29	28	simprd 113	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ⟨𝑏, 0R⟩ ∈ 𝐴)
30	22, 23, 29	rspcdva 2798	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ⟨𝑏, 0R⟩ <ℝ 𝑥)
31		simplrl 525	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → 𝑥 ∈ ℝ)
32	31, 19	sylib 121	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
33	32	simprd 113	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → 𝑥 = ⟨(1st ‘𝑥), 0R⟩)
34	30, 33	breqtrd 3962	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ⟨𝑏, 0R⟩ <ℝ ⟨(1st ‘𝑥), 0R⟩)
35		ltresr 7671	. . . . . . 7 ⊢ (⟨𝑏, 0R⟩ <ℝ ⟨(1st ‘𝑥), 0R⟩ ↔ 𝑏 <R (1st ‘𝑥))
36	34, 35	sylib 121	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → 𝑏 <R (1st ‘𝑥))
37	36	ralrimiva 2508	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ∀𝑏 ∈ 𝐵 𝑏 <R (1st ‘𝑥))
38		brralrspcev 3994	. . . . 5 ⊢ (((1st ‘𝑥) ∈ R ∧ ∀𝑏 ∈ 𝐵 𝑏 <R (1st ‘𝑥)) → ∃𝑎 ∈ R ∀𝑏 ∈ 𝐵 𝑏 <R 𝑎)
39	21, 37, 38	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ∃𝑎 ∈ R ∀𝑏 ∈ 𝐵 𝑏 <R 𝑎)
40	17, 39	rexlimddv 2557	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ R ∀𝑏 ∈ 𝐵 𝑏 <R 𝑎)
41		simpr 109	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → 𝑎 <R 𝑏)
42		ltresr 7671	. . . . . . . 8 ⊢ (⟨𝑎, 0R⟩ <ℝ ⟨𝑏, 0R⟩ ↔ 𝑎 <R 𝑏)
43	41, 42	sylibr 133	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ⟨𝑎, 0R⟩ <ℝ ⟨𝑏, 0R⟩)
44		breq2 3941	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → (⟨𝑎, 0R⟩ <ℝ 𝑦 ↔ ⟨𝑎, 0R⟩ <ℝ ⟨𝑏, 0R⟩))
45		breq2 3941	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → (𝑧 <ℝ 𝑦 ↔ 𝑧 <ℝ ⟨𝑏, 0R⟩))
46	45	ralbidv 2438	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → (∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩))
47	46	orbi2d 780	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → ((∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦) ↔ (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩)))
48	44, 47	imbi12d 233	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → ((⟨𝑎, 0R⟩ <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)) ↔ (⟨𝑎, 0R⟩ <ℝ ⟨𝑏, 0R⟩ → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩))))
49		breq1 3940	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (𝑥 <ℝ 𝑦 ↔ ⟨𝑎, 0R⟩ <ℝ 𝑦))
50		breq1 3940	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (𝑥 <ℝ 𝑧 ↔ ⟨𝑎, 0R⟩ <ℝ 𝑧))
51	50	rexbidv 2439	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ↔ ∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧))
52	51	orbi1d 781	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → ((∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦) ↔ (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
53	49, 52	imbi12d 233	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → ((𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)) ↔ (⟨𝑎, 0R⟩ <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦))))
54	53	ralbidv 2438	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)) ↔ ∀𝑦 ∈ ℝ (⟨𝑎, 0R⟩ <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦))))
55		axpre-suploclem.loc	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
56	55	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
57		simplrl 525	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → 𝑎 ∈ R)
58		opelreal 7659	. . . . . . . . . 10 ⊢ (⟨𝑎, 0R⟩ ∈ ℝ ↔ 𝑎 ∈ R)
59	57, 58	sylibr 133	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ⟨𝑎, 0R⟩ ∈ ℝ)
60	54, 56, 59	rspcdva 2798	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ∀𝑦 ∈ ℝ (⟨𝑎, 0R⟩ <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))
61		simplrr 526	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → 𝑏 ∈ R)
62		opelreal 7659	. . . . . . . . 9 ⊢ (⟨𝑏, 0R⟩ ∈ ℝ ↔ 𝑏 ∈ R)
63	61, 62	sylibr 133	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ⟨𝑏, 0R⟩ ∈ ℝ)
64	48, 60, 63	rspcdva 2798	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → (⟨𝑎, 0R⟩ <ℝ ⟨𝑏, 0R⟩ → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩)))
65	43, 64	mpd 13	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩))
66		simplll 523	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝜑)
67		simprl 521	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝑧 ∈ 𝐴)
68	1	sseld 3101	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ))
69	66, 67, 68	sylc 62	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝑧 ∈ ℝ)
70		elreal2 7662	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ ↔ ((1st ‘𝑧) ∈ R ∧ 𝑧 = ⟨(1st ‘𝑧), 0R⟩))
71	69, 70	sylib 121	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ((1st ‘𝑧) ∈ R ∧ 𝑧 = ⟨(1st ‘𝑧), 0R⟩))
72	71	simpld 111	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → (1st ‘𝑧) ∈ R)
73	71	simprd 113	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝑧 = ⟨(1st ‘𝑧), 0R⟩)
74	73, 67	eqeltrrd 2218	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ⟨(1st ‘𝑧), 0R⟩ ∈ 𝐴)
75		opeq1 3713	. . . . . . . . . . . 12 ⊢ (𝑤 = (1st ‘𝑧) → ⟨𝑤, 0R⟩ = ⟨(1st ‘𝑧), 0R⟩)
76	75	eleq1d 2209	. . . . . . . . . . 11 ⊢ (𝑤 = (1st ‘𝑧) → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨(1st ‘𝑧), 0R⟩ ∈ 𝐴))
77	76, 11	elrab2 2847	. . . . . . . . . 10 ⊢ ((1st ‘𝑧) ∈ 𝐵 ↔ ((1st ‘𝑧) ∈ R ∧ ⟨(1st ‘𝑧), 0R⟩ ∈ 𝐴))
78	72, 74, 77	sylanbrc 414	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → (1st ‘𝑧) ∈ 𝐵)
79		simprr 522	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ⟨𝑎, 0R⟩ <ℝ 𝑧)
80	79, 73	breqtrd 3962	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ⟨𝑎, 0R⟩ <ℝ ⟨(1st ‘𝑧), 0R⟩)
81		ltresr 7671	. . . . . . . . . 10 ⊢ (⟨𝑎, 0R⟩ <ℝ ⟨(1st ‘𝑧), 0R⟩ ↔ 𝑎 <R (1st ‘𝑧))
82	80, 81	sylib 121	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝑎 <R (1st ‘𝑧))
83		breq2 3941	. . . . . . . . . 10 ⊢ (𝑐 = (1st ‘𝑧) → (𝑎 <R 𝑐 ↔ 𝑎 <R (1st ‘𝑧)))
84	83	rspcev 2793	. . . . . . . . 9 ⊢ (((1st ‘𝑧) ∈ 𝐵 ∧ 𝑎 <R (1st ‘𝑧)) → ∃𝑐 ∈ 𝐵 𝑎 <R 𝑐)
85	78, 82, 84	syl2anc 409	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ∃𝑐 ∈ 𝐵 𝑎 <R 𝑐)
86	85	rexlimdvaa 2553	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 → ∃𝑐 ∈ 𝐵 𝑎 <R 𝑐))
87		breq1 3940	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑐, 0R⟩ → (𝑧 <ℝ ⟨𝑏, 0R⟩ ↔ ⟨𝑐, 0R⟩ <ℝ ⟨𝑏, 0R⟩))
88		simplr 520	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩)
89		simpr 109	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ 𝐵)
90		opeq1 3713	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑐 → ⟨𝑤, 0R⟩ = ⟨𝑐, 0R⟩)
91	90	eleq1d 2209	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑐 → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨𝑐, 0R⟩ ∈ 𝐴))
92	91, 11	elrab2 2847	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐵 ↔ (𝑐 ∈ R ∧ ⟨𝑐, 0R⟩ ∈ 𝐴))
93	89, 92	sylib 121	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → (𝑐 ∈ R ∧ ⟨𝑐, 0R⟩ ∈ 𝐴))
94	93	simprd 113	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → ⟨𝑐, 0R⟩ ∈ 𝐴)
95	87, 88, 94	rspcdva 2798	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → ⟨𝑐, 0R⟩ <ℝ ⟨𝑏, 0R⟩)
96		ltresr 7671	. . . . . . . . . 10 ⊢ (⟨𝑐, 0R⟩ <ℝ ⟨𝑏, 0R⟩ ↔ 𝑐 <R 𝑏)
97	95, 96	sylib 121	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → 𝑐 <R 𝑏)
98	97	ralrimiva 2508	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) → ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏)
99	98	ex 114	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → (∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩ → ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏))
100	86, 99	orim12d 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ((∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) → (∃𝑐 ∈ 𝐵 𝑎 <R 𝑐 ∨ ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏)))
101	65, 100	mpd 13	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → (∃𝑐 ∈ 𝐵 𝑎 <R 𝑐 ∨ ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏))
102	101	ex 114	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) → (𝑎 <R 𝑏 → (∃𝑐 ∈ 𝐵 𝑎 <R 𝑐 ∨ ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏)))
103	102	ralrimivva 2517	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ R ∀𝑏 ∈ R (𝑎 <R 𝑏 → (∃𝑐 ∈ 𝐵 𝑎 <R 𝑐 ∨ ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏)))
104	16, 40, 103	suplocsr 7641	. 2 ⊢ (𝜑 → ∃𝑎 ∈ R (∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))
105		simprl 521	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ (∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))) → 𝑎 ∈ R)
106	105, 58	sylibr 133	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ (∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))) → ⟨𝑎, 0R⟩ ∈ ℝ)
107		breq2 3941	. . . . . . . 8 ⊢ (𝑏 = (1st ‘𝑦) → (𝑎 <R 𝑏 ↔ 𝑎 <R (1st ‘𝑦)))
108	107	notbid 657	. . . . . . 7 ⊢ (𝑏 = (1st ‘𝑦) → (¬ 𝑎 <R 𝑏 ↔ ¬ 𝑎 <R (1st ‘𝑦)))
109		simplrr 526	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)
110	1	sselda 3102	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
111		elreal2 7662	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ ↔ ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
112	110, 111	sylib 121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
113	112	simpld 111	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ R)
114	112	simprd 113	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
115		simpr 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
116	114, 115	eqeltrrd 2218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⟨(1st ‘𝑦), 0R⟩ ∈ 𝐴)
117		opeq1 3713	. . . . . . . . . . 11 ⊢ (𝑤 = (1st ‘𝑦) → ⟨𝑤, 0R⟩ = ⟨(1st ‘𝑦), 0R⟩)
118	117	eleq1d 2209	. . . . . . . . . 10 ⊢ (𝑤 = (1st ‘𝑦) → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨(1st ‘𝑦), 0R⟩ ∈ 𝐴))
119	118, 11	elrab2 2847	. . . . . . . . 9 ⊢ ((1st ‘𝑦) ∈ 𝐵 ↔ ((1st ‘𝑦) ∈ R ∧ ⟨(1st ‘𝑦), 0R⟩ ∈ 𝐴))
120	113, 116, 119	sylanbrc 414	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ 𝐵)
121	120	adantlr 469	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ 𝐵)
122	108, 109, 121	rspcdva 2798	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → ¬ 𝑎 <R (1st ‘𝑦))
123	114	adantlr 469	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
124	123	breq2d 3949	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → (⟨𝑎, 0R⟩ <ℝ 𝑦 ↔ ⟨𝑎, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩))
125		ltresr 7671	. . . . . . 7 ⊢ (⟨𝑎, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩ ↔ 𝑎 <R (1st ‘𝑦))
126	124, 125	syl6bb 195	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → (⟨𝑎, 0R⟩ <ℝ 𝑦 ↔ 𝑎 <R (1st ‘𝑦)))
127	122, 126	mtbird 663	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦)
128	127	ralrimiva 2508	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦)
129	128	adantrrr 479	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ (∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))) → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦)
130		simplr 520	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → 𝑦 ∈ ℝ)
131	130, 111	sylib 121	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
132	131	simprd 113	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
133		simpr 109	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → 𝑦 <ℝ ⟨𝑎, 0R⟩)
134	132, 133	eqbrtrrd 3960	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → ⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑎, 0R⟩)
135		ltresr 7671	. . . . . . . . 9 ⊢ (⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑎, 0R⟩ ↔ (1st ‘𝑦) <R 𝑎)
136	134, 135	sylib 121	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → (1st ‘𝑦) <R 𝑎)
137		breq1 3940	. . . . . . . . . 10 ⊢ (𝑏 = (1st ‘𝑦) → (𝑏 <R 𝑎 ↔ (1st ‘𝑦) <R 𝑎))
138		breq1 3940	. . . . . . . . . . 11 ⊢ (𝑏 = (1st ‘𝑦) → (𝑏 <R 𝑐 ↔ (1st ‘𝑦) <R 𝑐))
139	138	rexbidv 2439	. . . . . . . . . 10 ⊢ (𝑏 = (1st ‘𝑦) → (∃𝑐 ∈ 𝐵 𝑏 <R 𝑐 ↔ ∃𝑐 ∈ 𝐵 (1st ‘𝑦) <R 𝑐))
140	137, 139	imbi12d 233	. . . . . . . . 9 ⊢ (𝑏 = (1st ‘𝑦) → ((𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐) ↔ ((1st ‘𝑦) <R 𝑎 → ∃𝑐 ∈ 𝐵 (1st ‘𝑦) <R 𝑐)))
141		simprr 522	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) → ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))
142	141	ad2antrr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))
143	131	simpld 111	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → (1st ‘𝑦) ∈ R)
144	140, 142, 143	rspcdva 2798	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → ((1st ‘𝑦) <R 𝑎 → ∃𝑐 ∈ 𝐵 (1st ‘𝑦) <R 𝑐))
145	136, 144	mpd 13	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → ∃𝑐 ∈ 𝐵 (1st ‘𝑦) <R 𝑐)
146		nfv 1509	. . . . . . . . . . 11 ⊢ Ⅎ𝑐𝜑
147		nfv 1509	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐 𝑎 ∈ R
148		nfcv 2282	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑐R
149		nfv 1509	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑐 𝑏 <R 𝑎
150		nfre1 2479	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑐∃𝑐 ∈ 𝐵 𝑏 <R 𝑐
151	149, 150	nfim 1552	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑐(𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)
152	148, 151	nfralya 2476	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)
153	147, 152	nfan 1545	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))
154	146, 153	nfan 1545	. . . . . . . . . 10 ⊢ Ⅎ𝑐(𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))
155		nfv 1509	. . . . . . . . . 10 ⊢ Ⅎ𝑐 𝑦 ∈ ℝ
156	154, 155	nfan 1545	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ)
157		nfv 1509	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑦 <ℝ ⟨𝑎, 0R⟩
158	156, 157	nfan 1545	. . . . . . . 8 ⊢ Ⅎ𝑐(((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩)
159		nfv 1509	. . . . . . . 8 ⊢ Ⅎ𝑐∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧
160		simprl 521	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) ∧ (𝑐 ∈ 𝐵 ∧ (1st ‘𝑦) <R 𝑐)) → 𝑐 ∈ 𝐵)
161	160, 92	sylib 121	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) ∧ (𝑐 ∈ 𝐵 ∧ (1st ‘𝑦) <R 𝑐)) → (𝑐 ∈ R ∧ ⟨𝑐, 0R⟩ ∈ 𝐴))
162	161	simprd 113	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) ∧ (𝑐 ∈ 𝐵 ∧ (1st ‘𝑦) <R 𝑐)) → ⟨𝑐, 0R⟩ ∈ 𝐴)
163	132	adantr 274	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) ∧ (𝑐 ∈ 𝐵 ∧ (1st ‘𝑦) <R 𝑐)) → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
164		simprr 522	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) ∧ (𝑐 ∈ 𝐵 ∧ (1st ‘𝑦) <R 𝑐)) → (1st ‘𝑦) <R 𝑐)
165		ltresr 7671	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑐, 0R⟩ ↔ (1st ‘𝑦) <R 𝑐)
166	164, 165	sylibr 133	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) ∧ (𝑐 ∈ 𝐵 ∧ (1st ‘𝑦) <R 𝑐)) → ⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑐, 0R⟩)
167	163, 166	eqbrtrd 3958	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) ∧ (𝑐 ∈ 𝐵 ∧ (1st ‘𝑦) <R 𝑐)) → 𝑦 <ℝ ⟨𝑐, 0R⟩)
168		breq2 3941	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑐, 0R⟩ → (𝑦 <ℝ 𝑧 ↔ 𝑦 <ℝ ⟨𝑐, 0R⟩))
169	168	rspcev 2793	. . . . . . . . . 10 ⊢ ((⟨𝑐, 0R⟩ ∈ 𝐴 ∧ 𝑦 <ℝ ⟨𝑐, 0R⟩) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)
170	162, 167, 169	syl2anc 409	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) ∧ (𝑐 ∈ 𝐵 ∧ (1st ‘𝑦) <R 𝑐)) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)
171	170	exp32 363	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → (𝑐 ∈ 𝐵 → ((1st ‘𝑦) <R 𝑐 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
172	158, 159, 171	rexlimd 2549	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → (∃𝑐 ∈ 𝐵 (1st ‘𝑦) <R 𝑐 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
173	145, 172	mpd 13	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)
174	173	ex 114	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) → (𝑦 <ℝ ⟨𝑎, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
175	174	ralrimiva 2508	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) → ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑎, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
176	175	adantrrl 478	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ (∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))) → ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑎, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
177	49	notbid 657	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (¬ 𝑥 <ℝ 𝑦 ↔ ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦))
178	177	ralbidv 2438	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦))
179		breq2 3941	. . . . . . 7 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (𝑦 <ℝ 𝑥 ↔ 𝑦 <ℝ ⟨𝑎, 0R⟩))
180	179	imbi1d 230	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → ((𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ (𝑦 <ℝ ⟨𝑎, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
181	180	ralbidv 2438	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑎, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
182	178, 181	anbi12d 465	. . . 4 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑎, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
183	182	rspcev 2793	. . 3 ⊢ ((⟨𝑎, 0R⟩ ∈ ℝ ∧ (∀𝑦 ∈ 𝐴 ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑎, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
184	106, 129, 176, 183	syl12anc 1215	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ (∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
185	104, 184	rexlimddv 2557	1 ⊢ (𝜑 → ∃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 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  {crab 2421   ⊆ wss 3076  ⟨cop 3535   class class class wbr 3937  ‘cfv 5131  1^st c1st 6044  Rcnr 7129  0_Rc0r 7130   <_R cltr 7135  ℝcr 7643   <_ℝ cltrr 7648

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-i1p 7299  df-iplp 7300  df-imp 7301  df-iltp 7302  df-enr 7558  df-nr 7559  df-plr 7560  df-mr 7561  df-ltr 7562  df-0r 7563  df-1r 7564  df-m1r 7565  df-r 7654  df-lt 7657

This theorem is referenced by:  axpre-suploc  7734