Theorem axpre-suploclemres 7891

Description: Lemma for axpre-suploc 7892. 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 7799. (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 3156	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4		elreal2 7820	. . . . . . 7 ⊢ (𝐶 ∈ ℝ ↔ ((1st ‘𝐶) ∈ R ∧ 𝐶 = ⟨(1st ‘𝐶), 0R⟩))
5	3, 4	sylib 122	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐶) ∈ R ∧ 𝐶 = ⟨(1st ‘𝐶), 0R⟩))
6	5	simpld 112	. . . . 5 ⊢ (𝜑 → (1st ‘𝐶) ∈ R)
7	5	simprd 114	. . . . . 6 ⊢ (𝜑 → 𝐶 = ⟨(1st ‘𝐶), 0R⟩)
8	7, 2	eqeltrrd 2255	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘𝐶), 0R⟩ ∈ 𝐴)
9		opeq1 3776	. . . . . . 7 ⊢ (𝑤 = (1st ‘𝐶) → ⟨𝑤, 0R⟩ = ⟨(1st ‘𝐶), 0R⟩)
10	9	eleq1d 2246	. . . . . 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 2896	. . . . 5 ⊢ ((1st ‘𝐶) ∈ 𝐵 ↔ ((1st ‘𝐶) ∈ R ∧ ⟨(1st ‘𝐶), 0R⟩ ∈ 𝐴))
13	6, 8, 12	sylanbrc 417	. . . 4 ⊢ (𝜑 → (1st ‘𝐶) ∈ 𝐵)
14		eleq1 2240	. . . . 5 ⊢ (𝑎 = (1st ‘𝐶) → (𝑎 ∈ 𝐵 ↔ (1st ‘𝐶) ∈ 𝐵))
15	14	spcegv 2825	. . . 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 529	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → 𝑥 ∈ ℝ)
19		elreal2 7820	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↔ ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
20	18, 19	sylib 122	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
21	20	simpld 112	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (1st ‘𝑥) ∈ R)
22		breq1 4003	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → (𝑦 <ℝ 𝑥 ↔ ⟨𝑏, 0R⟩ <ℝ 𝑥))
23		simplrr 536	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)
24		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
25		opeq1 3776	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑏 → ⟨𝑤, 0R⟩ = ⟨𝑏, 0R⟩)
26	25	eleq1d 2246	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑏 → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨𝑏, 0R⟩ ∈ 𝐴))
27	26, 11	elrab2 2896	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ R ∧ ⟨𝑏, 0R⟩ ∈ 𝐴))
28	24, 27	sylib 122	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → (𝑏 ∈ R ∧ ⟨𝑏, 0R⟩ ∈ 𝐴))
29	28	simprd 114	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ⟨𝑏, 0R⟩ ∈ 𝐴)
30	22, 23, 29	rspcdva 2846	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ⟨𝑏, 0R⟩ <ℝ 𝑥)
31		simplrl 535	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → 𝑥 ∈ ℝ)
32	31, 19	sylib 122	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
33	32	simprd 114	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → 𝑥 = ⟨(1st ‘𝑥), 0R⟩)
34	30, 33	breqtrd 4026	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → ⟨𝑏, 0R⟩ <ℝ ⟨(1st ‘𝑥), 0R⟩)
35		ltresr 7829	. . . . . . 7 ⊢ (⟨𝑏, 0R⟩ <ℝ ⟨(1st ‘𝑥), 0R⟩ ↔ 𝑏 <R (1st ‘𝑥))
36	34, 35	sylib 122	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) ∧ 𝑏 ∈ 𝐵) → 𝑏 <R (1st ‘𝑥))
37	36	ralrimiva 2550	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ∀𝑏 ∈ 𝐵 𝑏 <R (1st ‘𝑥))
38		brralrspcev 4058	. . . . 5 ⊢ (((1st ‘𝑥) ∈ R ∧ ∀𝑏 ∈ 𝐵 𝑏 <R (1st ‘𝑥)) → ∃𝑎 ∈ R ∀𝑏 ∈ 𝐵 𝑏 <R 𝑎)
39	21, 37, 38	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ∃𝑎 ∈ R ∀𝑏 ∈ 𝐵 𝑏 <R 𝑎)
40	17, 39	rexlimddv 2599	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ R ∀𝑏 ∈ 𝐵 𝑏 <R 𝑎)
41		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → 𝑎 <R 𝑏)
42		ltresr 7829	. . . . . . . 8 ⊢ (⟨𝑎, 0R⟩ <ℝ ⟨𝑏, 0R⟩ ↔ 𝑎 <R 𝑏)
43	41, 42	sylibr 134	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ⟨𝑎, 0R⟩ <ℝ ⟨𝑏, 0R⟩)
44		breq2 4004	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → (⟨𝑎, 0R⟩ <ℝ 𝑦 ↔ ⟨𝑎, 0R⟩ <ℝ ⟨𝑏, 0R⟩))
45		breq2 4004	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → (𝑧 <ℝ 𝑦 ↔ 𝑧 <ℝ ⟨𝑏, 0R⟩))
46	45	ralbidv 2477	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑏, 0R⟩ → (∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩))
47	46	orbi2d 790	. . . . . . . . 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 234	. . . . . . . 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 4003	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (𝑥 <ℝ 𝑦 ↔ ⟨𝑎, 0R⟩ <ℝ 𝑦))
50		breq1 4003	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (𝑥 <ℝ 𝑧 ↔ ⟨𝑎, 0R⟩ <ℝ 𝑧))
51	50	rexbidv 2478	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ↔ ∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧))
52	51	orbi1d 791	. . . . . . . . . . 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 234	. . . . . . . . . 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 2477	. . . . . . . . 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 488	. . . . . . . . 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 535	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → 𝑎 ∈ R)
58		opelreal 7817	. . . . . . . . . 10 ⊢ (⟨𝑎, 0R⟩ ∈ ℝ ↔ 𝑎 ∈ R)
59	57, 58	sylibr 134	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ⟨𝑎, 0R⟩ ∈ ℝ)
60	54, 56, 59	rspcdva 2846	. . . . . . . 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 536	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → 𝑏 ∈ R)
62		opelreal 7817	. . . . . . . . 9 ⊢ (⟨𝑏, 0R⟩ ∈ ℝ ↔ 𝑏 ∈ R)
63	61, 62	sylibr 134	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → ⟨𝑏, 0R⟩ ∈ ℝ)
64	48, 60, 63	rspcdva 2846	. . . . . . 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 533	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝜑)
67		simprl 529	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝑧 ∈ 𝐴)
68	1	sseld 3154	. . . . . . . . . . . . 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 7820	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ ↔ ((1st ‘𝑧) ∈ R ∧ 𝑧 = ⟨(1st ‘𝑧), 0R⟩))
71	69, 70	sylib 122	. . . . . . . . . . 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 112	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → (1st ‘𝑧) ∈ R)
73	71	simprd 114	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝑧 = ⟨(1st ‘𝑧), 0R⟩)
74	73, 67	eqeltrrd 2255	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ⟨(1st ‘𝑧), 0R⟩ ∈ 𝐴)
75		opeq1 3776	. . . . . . . . . . . 12 ⊢ (𝑤 = (1st ‘𝑧) → ⟨𝑤, 0R⟩ = ⟨(1st ‘𝑧), 0R⟩)
76	75	eleq1d 2246	. . . . . . . . . . 11 ⊢ (𝑤 = (1st ‘𝑧) → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨(1st ‘𝑧), 0R⟩ ∈ 𝐴))
77	76, 11	elrab2 2896	. . . . . . . . . 10 ⊢ ((1st ‘𝑧) ∈ 𝐵 ↔ ((1st ‘𝑧) ∈ R ∧ ⟨(1st ‘𝑧), 0R⟩ ∈ 𝐴))
78	72, 74, 77	sylanbrc 417	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → (1st ‘𝑧) ∈ 𝐵)
79		simprr 531	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ⟨𝑎, 0R⟩ <ℝ 𝑧)
80	79, 73	breqtrd 4026	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ⟨𝑎, 0R⟩ <ℝ ⟨(1st ‘𝑧), 0R⟩)
81		ltresr 7829	. . . . . . . . . 10 ⊢ (⟨𝑎, 0R⟩ <ℝ ⟨(1st ‘𝑧), 0R⟩ ↔ 𝑎 <R (1st ‘𝑧))
82	80, 81	sylib 122	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → 𝑎 <R (1st ‘𝑧))
83		breq2 4004	. . . . . . . . . 10 ⊢ (𝑐 = (1st ‘𝑧) → (𝑎 <R 𝑐 ↔ 𝑎 <R (1st ‘𝑧)))
84	83	rspcev 2841	. . . . . . . . 9 ⊢ (((1st ‘𝑧) ∈ 𝐵 ∧ 𝑎 <R (1st ‘𝑧)) → ∃𝑐 ∈ 𝐵 𝑎 <R 𝑐)
85	78, 82, 84	syl2anc 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧 ∈ 𝐴 ∧ ⟨𝑎, 0R⟩ <ℝ 𝑧)) → ∃𝑐 ∈ 𝐵 𝑎 <R 𝑐)
86	85	rexlimdvaa 2595	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → (∃𝑧 ∈ 𝐴 ⟨𝑎, 0R⟩ <ℝ 𝑧 → ∃𝑐 ∈ 𝐵 𝑎 <R 𝑐))
87		breq1 4003	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑐, 0R⟩ → (𝑧 <ℝ ⟨𝑏, 0R⟩ ↔ ⟨𝑐, 0R⟩ <ℝ ⟨𝑏, 0R⟩))
88		simplr 528	. . . . . . . . . . 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 110	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ 𝐵)
90		opeq1 3776	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑐 → ⟨𝑤, 0R⟩ = ⟨𝑐, 0R⟩)
91	90	eleq1d 2246	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑐 → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨𝑐, 0R⟩ ∈ 𝐴))
92	91, 11	elrab2 2896	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐵 ↔ (𝑐 ∈ R ∧ ⟨𝑐, 0R⟩ ∈ 𝐴))
93	89, 92	sylib 122	. . . . . . . . . . . 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 114	. . . . . . . . . . 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 2846	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → ⟨𝑐, 0R⟩ <ℝ ⟨𝑏, 0R⟩)
96		ltresr 7829	. . . . . . . . . 10 ⊢ (⟨𝑐, 0R⟩ <ℝ ⟨𝑏, 0R⟩ ↔ 𝑐 <R 𝑏)
97	95, 96	sylib 122	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) ∧ 𝑐 ∈ 𝐵) → 𝑐 <R 𝑏)
98	97	ralrimiva 2550	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩) → ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏)
99	98	ex 115	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) ∧ 𝑎 <R 𝑏) → (∀𝑧 ∈ 𝐴 𝑧 <ℝ ⟨𝑏, 0R⟩ → ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏))
100	86, 99	orim12d 786	. . . . . 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 115	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ 𝑏 ∈ R)) → (𝑎 <R 𝑏 → (∃𝑐 ∈ 𝐵 𝑎 <R 𝑐 ∨ ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏)))
103	102	ralrimivva 2559	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ R ∀𝑏 ∈ R (𝑎 <R 𝑏 → (∃𝑐 ∈ 𝐵 𝑎 <R 𝑐 ∨ ∀𝑐 ∈ 𝐵 𝑐 <R 𝑏)))
104	16, 40, 103	suplocsr 7799	. 2 ⊢ (𝜑 → ∃𝑎 ∈ R (∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))
105		simprl 529	. . . 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 134	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ (∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))) → ⟨𝑎, 0R⟩ ∈ ℝ)
107		breq2 4004	. . . . . . . 8 ⊢ (𝑏 = (1st ‘𝑦) → (𝑎 <R 𝑏 ↔ 𝑎 <R (1st ‘𝑦)))
108	107	notbid 667	. . . . . . 7 ⊢ (𝑏 = (1st ‘𝑦) → (¬ 𝑎 <R 𝑏 ↔ ¬ 𝑎 <R (1st ‘𝑦)))
109		simplrr 536	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)
110	1	sselda 3155	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
111		elreal2 7820	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ ↔ ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
112	110, 111	sylib 122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
113	112	simpld 112	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ R)
114	112	simprd 114	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
115		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
116	114, 115	eqeltrrd 2255	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⟨(1st ‘𝑦), 0R⟩ ∈ 𝐴)
117		opeq1 3776	. . . . . . . . . . 11 ⊢ (𝑤 = (1st ‘𝑦) → ⟨𝑤, 0R⟩ = ⟨(1st ‘𝑦), 0R⟩)
118	117	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑤 = (1st ‘𝑦) → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨(1st ‘𝑦), 0R⟩ ∈ 𝐴))
119	118, 11	elrab2 2896	. . . . . . . . 9 ⊢ ((1st ‘𝑦) ∈ 𝐵 ↔ ((1st ‘𝑦) ∈ R ∧ ⟨(1st ‘𝑦), 0R⟩ ∈ 𝐴))
120	113, 116, 119	sylanbrc 417	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ 𝐵)
121	120	adantlr 477	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ 𝐵)
122	108, 109, 121	rspcdva 2846	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → ¬ 𝑎 <R (1st ‘𝑦))
123	114	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
124	123	breq2d 4012	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → (⟨𝑎, 0R⟩ <ℝ 𝑦 ↔ ⟨𝑎, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩))
125		ltresr 7829	. . . . . . 7 ⊢ (⟨𝑎, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩ ↔ 𝑎 <R (1st ‘𝑦))
126	124, 125	bitrdi 196	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → (⟨𝑎, 0R⟩ <ℝ 𝑦 ↔ 𝑎 <R (1st ‘𝑦)))
127	122, 126	mtbird 673	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦 ∈ 𝐴) → ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦)
128	127	ralrimiva 2550	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑎 <R 𝑏)) → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦)
129	128	adantrrr 487	. . 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 528	. . . . . . . . . . . 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 122	. . . . . . . . . . 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 114	. . . . . . . . . 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 110	. . . . . . . . . 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 4024	. . . . . . . . 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 7829	. . . . . . . . 9 ⊢ (⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑎, 0R⟩ ↔ (1st ‘𝑦) <R 𝑎)
136	134, 135	sylib 122	. . . . . . . 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 4003	. . . . . . . . . 10 ⊢ (𝑏 = (1st ‘𝑦) → (𝑏 <R 𝑎 ↔ (1st ‘𝑦) <R 𝑎))
138		breq1 4003	. . . . . . . . . . 11 ⊢ (𝑏 = (1st ‘𝑦) → (𝑏 <R 𝑐 ↔ (1st ‘𝑦) <R 𝑐))
139	138	rexbidv 2478	. . . . . . . . . 10 ⊢ (𝑏 = (1st ‘𝑦) → (∃𝑐 ∈ 𝐵 𝑏 <R 𝑐 ↔ ∃𝑐 ∈ 𝐵 (1st ‘𝑦) <R 𝑐))
140	137, 139	imbi12d 234	. . . . . . . . 9 ⊢ (𝑏 = (1st ‘𝑦) → ((𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐) ↔ ((1st ‘𝑦) <R 𝑎 → ∃𝑐 ∈ 𝐵 (1st ‘𝑦) <R 𝑐)))
141		simprr 531	. . . . . . . . . 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 488	. . . . . . . . 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 112	. . . . . . . . 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 2846	. . . . . . . 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 1528	. . . . . . . . . . 11 ⊢ Ⅎ𝑐𝜑
147		nfv 1528	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐 𝑎 ∈ R
148		nfcv 2319	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑐R
149		nfv 1528	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑐 𝑏 <R 𝑎
150		nfre1 2520	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑐∃𝑐 ∈ 𝐵 𝑏 <R 𝑐
151	149, 150	nfim 1572	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑐(𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)
152	148, 151	nfralya 2517	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)
153	147, 152	nfan 1565	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))
154	146, 153	nfan 1565	. . . . . . . . . 10 ⊢ Ⅎ𝑐(𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐)))
155		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑐 𝑦 ∈ ℝ
156	154, 155	nfan 1565	. . . . . . . . 9 ⊢ Ⅎ𝑐((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ)
157		nfv 1528	. . . . . . . . 9 ⊢ Ⅎ𝑐 𝑦 <ℝ ⟨𝑎, 0R⟩
158	156, 157	nfan 1565	. . . . . . . 8 ⊢ Ⅎ𝑐(((𝜑 ∧ (𝑎 ∈ R ∧ ∀𝑏 ∈ R (𝑏 <R 𝑎 → ∃𝑐 ∈ 𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <ℝ ⟨𝑎, 0R⟩)
159		nfv 1528	. . . . . . . 8 ⊢ Ⅎ𝑐∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧
160		simprl 529	. . . . . . . . . . . 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 122	. . . . . . . . . . 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 114	. . . . . . . . . 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 276	. . . . . . . . . . 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 531	. . . . . . . . . . . 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 7829	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑐, 0R⟩ ↔ (1st ‘𝑦) <R 𝑐)
166	164, 165	sylibr 134	. . . . . . . . . . 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 4022	. . . . . . . . . 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 4004	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑐, 0R⟩ → (𝑦 <ℝ 𝑧 ↔ 𝑦 <ℝ ⟨𝑐, 0R⟩))
169	168	rspcev 2841	. . . . . . . . . 10 ⊢ ((⟨𝑐, 0R⟩ ∈ 𝐴 ∧ 𝑦 <ℝ ⟨𝑐, 0R⟩) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)
170	162, 167, 169	syl2anc 411	. . . . . . . . 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 365	. . . . . . . 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 2591	. . . . . . 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 115	. . . . 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 2550	. . . 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 486	. . 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 667	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (¬ 𝑥 <ℝ 𝑦 ↔ ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦))
178	177	ralbidv 2477	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ⟨𝑎, 0R⟩ <ℝ 𝑦))
179		breq2 4004	. . . . . . 7 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → (𝑦 <ℝ 𝑥 ↔ 𝑦 <ℝ ⟨𝑎, 0R⟩))
180	179	imbi1d 231	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 0R⟩ → ((𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ (𝑦 <ℝ ⟨𝑎, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
181	180	ralbidv 2477	. . . . 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 473	. . . 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 2841	. . 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 1236	. 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 2599	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 104   ∨ wo 708   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459   ⊆ wss 3129  ⟨cop 3594   class class class wbr 4000  ‘cfv 5212  1^st c1st 6133  Rcnr 7287  0_Rc0r 7288   <_R cltr 7293  ℝcr 7801   <_ℝ cltrr 7806

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-imp 7459  df-iltp 7460  df-enr 7716  df-nr 7717  df-plr 7718  df-mr 7719  df-ltr 7720  df-0r 7721  df-1r 7722  df-m1r 7723  df-r 7812  df-lt 7815

This theorem is referenced by:  axpre-suploc  7892