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Theorem axpre-suploclemres 7702
Description: Lemma for axpre-suploc 7703. 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 ), and apply suplocsr 7610. (Contributed by Jim Kingdon, 24-Jan-2024.)
Hypotheses
Ref Expression
axpre-suploclem.ss (𝜑𝐴 ⊆ ℝ)
axpre-suploclem.m (𝜑𝐶𝐴)
axpre-suploclem.ub (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥)
axpre-suploclem.loc (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))
axpre-suploclem.b 𝐵 = {𝑤R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}
Assertion
Ref Expression
axpre-suploclemres (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 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.
StepHypRef Expression
1 axpre-suploclem.ss . . . . . . . 8 (𝜑𝐴 ⊆ ℝ)
2 axpre-suploclem.m . . . . . . . 8 (𝜑𝐶𝐴)
31, 2sseldd 3093 . . . . . . 7 (𝜑𝐶 ∈ ℝ)
4 elreal2 7631 . . . . . . 7 (𝐶 ∈ ℝ ↔ ((1st𝐶) ∈ R𝐶 = ⟨(1st𝐶), 0R⟩))
53, 4sylib 121 . . . . . 6 (𝜑 → ((1st𝐶) ∈ R𝐶 = ⟨(1st𝐶), 0R⟩))
65simpld 111 . . . . 5 (𝜑 → (1st𝐶) ∈ R)
75simprd 113 . . . . . 6 (𝜑𝐶 = ⟨(1st𝐶), 0R⟩)
87, 2eqeltrrd 2215 . . . . 5 (𝜑 → ⟨(1st𝐶), 0R⟩ ∈ 𝐴)
9 opeq1 3700 . . . . . . 7 (𝑤 = (1st𝐶) → ⟨𝑤, 0R⟩ = ⟨(1st𝐶), 0R⟩)
109eleq1d 2206 . . . . . 6 (𝑤 = (1st𝐶) → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨(1st𝐶), 0R⟩ ∈ 𝐴))
11 axpre-suploclem.b . . . . . 6 𝐵 = {𝑤R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}
1210, 11elrab2 2838 . . . . 5 ((1st𝐶) ∈ 𝐵 ↔ ((1st𝐶) ∈ R ∧ ⟨(1st𝐶), 0R⟩ ∈ 𝐴))
136, 8, 12sylanbrc 413 . . . 4 (𝜑 → (1st𝐶) ∈ 𝐵)
14 eleq1 2200 . . . . 5 (𝑎 = (1st𝐶) → (𝑎𝐵 ↔ (1st𝐶) ∈ 𝐵))
1514spcegv 2769 . . . 4 ((1st𝐶) ∈ 𝐵 → ((1st𝐶) ∈ 𝐵 → ∃𝑎 𝑎𝐵))
1613, 13, 15sylc 62 . . 3 (𝜑 → ∃𝑎 𝑎𝐵)
17 axpre-suploclem.ub . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥)
18 simprl 520 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → 𝑥 ∈ ℝ)
19 elreal2 7631 . . . . . . 7 (𝑥 ∈ ℝ ↔ ((1st𝑥) ∈ R𝑥 = ⟨(1st𝑥), 0R⟩))
2018, 19sylib 121 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → ((1st𝑥) ∈ R𝑥 = ⟨(1st𝑥), 0R⟩))
2120simpld 111 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → (1st𝑥) ∈ R)
22 breq1 3927 . . . . . . . . 9 (𝑦 = ⟨𝑏, 0R⟩ → (𝑦 < 𝑥 ↔ ⟨𝑏, 0R⟩ < 𝑥))
23 simplrr 525 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → ∀𝑦𝐴 𝑦 < 𝑥)
24 simpr 109 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → 𝑏𝐵)
25 opeq1 3700 . . . . . . . . . . . . 13 (𝑤 = 𝑏 → ⟨𝑤, 0R⟩ = ⟨𝑏, 0R⟩)
2625eleq1d 2206 . . . . . . . . . . . 12 (𝑤 = 𝑏 → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨𝑏, 0R⟩ ∈ 𝐴))
2726, 11elrab2 2838 . . . . . . . . . . 11 (𝑏𝐵 ↔ (𝑏R ∧ ⟨𝑏, 0R⟩ ∈ 𝐴))
2824, 27sylib 121 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → (𝑏R ∧ ⟨𝑏, 0R⟩ ∈ 𝐴))
2928simprd 113 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → ⟨𝑏, 0R⟩ ∈ 𝐴)
3022, 23, 29rspcdva 2789 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → ⟨𝑏, 0R⟩ < 𝑥)
31 simplrl 524 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → 𝑥 ∈ ℝ)
3231, 19sylib 121 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → ((1st𝑥) ∈ R𝑥 = ⟨(1st𝑥), 0R⟩))
3332simprd 113 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → 𝑥 = ⟨(1st𝑥), 0R⟩)
3430, 33breqtrd 3949 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → ⟨𝑏, 0R⟩ < ⟨(1st𝑥), 0R⟩)
35 ltresr 7640 . . . . . . 7 (⟨𝑏, 0R⟩ < ⟨(1st𝑥), 0R⟩ ↔ 𝑏 <R (1st𝑥))
3634, 35sylib 121 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) ∧ 𝑏𝐵) → 𝑏 <R (1st𝑥))
3736ralrimiva 2503 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → ∀𝑏𝐵 𝑏 <R (1st𝑥))
38 brralrspcev 3981 . . . . 5 (((1st𝑥) ∈ R ∧ ∀𝑏𝐵 𝑏 <R (1st𝑥)) → ∃𝑎R𝑏𝐵 𝑏 <R 𝑎)
3921, 37, 38syl2anc 408 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → ∃𝑎R𝑏𝐵 𝑏 <R 𝑎)
4017, 39rexlimddv 2552 . . 3 (𝜑 → ∃𝑎R𝑏𝐵 𝑏 <R 𝑎)
41 simpr 109 . . . . . . . 8 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → 𝑎 <R 𝑏)
42 ltresr 7640 . . . . . . . 8 (⟨𝑎, 0R⟩ <𝑏, 0R⟩ ↔ 𝑎 <R 𝑏)
4341, 42sylibr 133 . . . . . . 7 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → ⟨𝑎, 0R⟩ <𝑏, 0R⟩)
44 breq2 3928 . . . . . . . . 9 (𝑦 = ⟨𝑏, 0R⟩ → (⟨𝑎, 0R⟩ < 𝑦 ↔ ⟨𝑎, 0R⟩ <𝑏, 0R⟩))
45 breq2 3928 . . . . . . . . . . 11 (𝑦 = ⟨𝑏, 0R⟩ → (𝑧 < 𝑦𝑧 <𝑏, 0R⟩))
4645ralbidv 2435 . . . . . . . . . 10 (𝑦 = ⟨𝑏, 0R⟩ → (∀𝑧𝐴 𝑧 < 𝑦 ↔ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩))
4746orbi2d 779 . . . . . . . . 9 (𝑦 = ⟨𝑏, 0R⟩ → ((∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦) ↔ (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩)))
4844, 47imbi12d 233 . . . . . . . 8 (𝑦 = ⟨𝑏, 0R⟩ → ((⟨𝑎, 0R⟩ < 𝑦 → (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)) ↔ (⟨𝑎, 0R⟩ <𝑏, 0R⟩ → (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩))))
49 breq1 3927 . . . . . . . . . . 11 (𝑥 = ⟨𝑎, 0R⟩ → (𝑥 < 𝑦 ↔ ⟨𝑎, 0R⟩ < 𝑦))
50 breq1 3927 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑎, 0R⟩ → (𝑥 < 𝑧 ↔ ⟨𝑎, 0R⟩ < 𝑧))
5150rexbidv 2436 . . . . . . . . . . . 12 (𝑥 = ⟨𝑎, 0R⟩ → (∃𝑧𝐴 𝑥 < 𝑧 ↔ ∃𝑧𝐴𝑎, 0R⟩ < 𝑧))
5251orbi1d 780 . . . . . . . . . . 11 (𝑥 = ⟨𝑎, 0R⟩ → ((∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦) ↔ (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))
5349, 52imbi12d 233 . . . . . . . . . 10 (𝑥 = ⟨𝑎, 0R⟩ → ((𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)) ↔ (⟨𝑎, 0R⟩ < 𝑦 → (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))))
5453ralbidv 2435 . . . . . . . . 9 (𝑥 = ⟨𝑎, 0R⟩ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)) ↔ ∀𝑦 ∈ ℝ (⟨𝑎, 0R⟩ < 𝑦 → (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))))
55 axpre-suploclem.loc . . . . . . . . . 10 (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))
5655ad2antrr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))
57 simplrl 524 . . . . . . . . . 10 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → 𝑎R)
58 opelreal 7628 . . . . . . . . . 10 (⟨𝑎, 0R⟩ ∈ ℝ ↔ 𝑎R)
5957, 58sylibr 133 . . . . . . . . 9 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → ⟨𝑎, 0R⟩ ∈ ℝ)
6054, 56, 59rspcdva 2789 . . . . . . . 8 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → ∀𝑦 ∈ ℝ (⟨𝑎, 0R⟩ < 𝑦 → (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))
61 simplrr 525 . . . . . . . . 9 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → 𝑏R)
62 opelreal 7628 . . . . . . . . 9 (⟨𝑏, 0R⟩ ∈ ℝ ↔ 𝑏R)
6361, 62sylibr 133 . . . . . . . 8 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → ⟨𝑏, 0R⟩ ∈ ℝ)
6448, 60, 63rspcdva 2789 . . . . . . 7 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → (⟨𝑎, 0R⟩ <𝑏, 0R⟩ → (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩)))
6543, 64mpd 13 . . . . . 6 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩))
66 simplll 522 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → 𝜑)
67 simprl 520 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → 𝑧𝐴)
681sseld 3091 . . . . . . . . . . . . 13 (𝜑 → (𝑧𝐴𝑧 ∈ ℝ))
6966, 67, 68sylc 62 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → 𝑧 ∈ ℝ)
70 elreal2 7631 . . . . . . . . . . . 12 (𝑧 ∈ ℝ ↔ ((1st𝑧) ∈ R𝑧 = ⟨(1st𝑧), 0R⟩))
7169, 70sylib 121 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → ((1st𝑧) ∈ R𝑧 = ⟨(1st𝑧), 0R⟩))
7271simpld 111 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → (1st𝑧) ∈ R)
7371simprd 113 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → 𝑧 = ⟨(1st𝑧), 0R⟩)
7473, 67eqeltrrd 2215 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → ⟨(1st𝑧), 0R⟩ ∈ 𝐴)
75 opeq1 3700 . . . . . . . . . . . 12 (𝑤 = (1st𝑧) → ⟨𝑤, 0R⟩ = ⟨(1st𝑧), 0R⟩)
7675eleq1d 2206 . . . . . . . . . . 11 (𝑤 = (1st𝑧) → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨(1st𝑧), 0R⟩ ∈ 𝐴))
7776, 11elrab2 2838 . . . . . . . . . 10 ((1st𝑧) ∈ 𝐵 ↔ ((1st𝑧) ∈ R ∧ ⟨(1st𝑧), 0R⟩ ∈ 𝐴))
7872, 74, 77sylanbrc 413 . . . . . . . . 9 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → (1st𝑧) ∈ 𝐵)
79 simprr 521 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → ⟨𝑎, 0R⟩ < 𝑧)
8079, 73breqtrd 3949 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → ⟨𝑎, 0R⟩ < ⟨(1st𝑧), 0R⟩)
81 ltresr 7640 . . . . . . . . . 10 (⟨𝑎, 0R⟩ < ⟨(1st𝑧), 0R⟩ ↔ 𝑎 <R (1st𝑧))
8280, 81sylib 121 . . . . . . . . 9 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → 𝑎 <R (1st𝑧))
83 breq2 3928 . . . . . . . . . 10 (𝑐 = (1st𝑧) → (𝑎 <R 𝑐𝑎 <R (1st𝑧)))
8483rspcev 2784 . . . . . . . . 9 (((1st𝑧) ∈ 𝐵𝑎 <R (1st𝑧)) → ∃𝑐𝐵 𝑎 <R 𝑐)
8578, 82, 84syl2anc 408 . . . . . . . 8 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ (𝑧𝐴 ∧ ⟨𝑎, 0R⟩ < 𝑧)) → ∃𝑐𝐵 𝑎 <R 𝑐)
8685rexlimdvaa 2548 . . . . . . 7 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → (∃𝑧𝐴𝑎, 0R⟩ < 𝑧 → ∃𝑐𝐵 𝑎 <R 𝑐))
87 breq1 3927 . . . . . . . . . . 11 (𝑧 = ⟨𝑐, 0R⟩ → (𝑧 <𝑏, 0R⟩ ↔ ⟨𝑐, 0R⟩ <𝑏, 0R⟩))
88 simplr 519 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩) ∧ 𝑐𝐵) → ∀𝑧𝐴 𝑧 <𝑏, 0R⟩)
89 simpr 109 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩) ∧ 𝑐𝐵) → 𝑐𝐵)
90 opeq1 3700 . . . . . . . . . . . . . . 15 (𝑤 = 𝑐 → ⟨𝑤, 0R⟩ = ⟨𝑐, 0R⟩)
9190eleq1d 2206 . . . . . . . . . . . . . 14 (𝑤 = 𝑐 → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨𝑐, 0R⟩ ∈ 𝐴))
9291, 11elrab2 2838 . . . . . . . . . . . . 13 (𝑐𝐵 ↔ (𝑐R ∧ ⟨𝑐, 0R⟩ ∈ 𝐴))
9389, 92sylib 121 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩) ∧ 𝑐𝐵) → (𝑐R ∧ ⟨𝑐, 0R⟩ ∈ 𝐴))
9493simprd 113 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩) ∧ 𝑐𝐵) → ⟨𝑐, 0R⟩ ∈ 𝐴)
9587, 88, 94rspcdva 2789 . . . . . . . . . 10 (((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩) ∧ 𝑐𝐵) → ⟨𝑐, 0R⟩ <𝑏, 0R⟩)
96 ltresr 7640 . . . . . . . . . 10 (⟨𝑐, 0R⟩ <𝑏, 0R⟩ ↔ 𝑐 <R 𝑏)
9795, 96sylib 121 . . . . . . . . 9 (((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩) ∧ 𝑐𝐵) → 𝑐 <R 𝑏)
9897ralrimiva 2503 . . . . . . . 8 ((((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) ∧ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩) → ∀𝑐𝐵 𝑐 <R 𝑏)
9998ex 114 . . . . . . 7 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → (∀𝑧𝐴 𝑧 <𝑏, 0R⟩ → ∀𝑐𝐵 𝑐 <R 𝑏))
10086, 99orim12d 775 . . . . . 6 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → ((∃𝑧𝐴𝑎, 0R⟩ < 𝑧 ∨ ∀𝑧𝐴 𝑧 <𝑏, 0R⟩) → (∃𝑐𝐵 𝑎 <R 𝑐 ∨ ∀𝑐𝐵 𝑐 <R 𝑏)))
10165, 100mpd 13 . . . . 5 (((𝜑 ∧ (𝑎R𝑏R)) ∧ 𝑎 <R 𝑏) → (∃𝑐𝐵 𝑎 <R 𝑐 ∨ ∀𝑐𝐵 𝑐 <R 𝑏))
102101ex 114 . . . 4 ((𝜑 ∧ (𝑎R𝑏R)) → (𝑎 <R 𝑏 → (∃𝑐𝐵 𝑎 <R 𝑐 ∨ ∀𝑐𝐵 𝑐 <R 𝑏)))
103102ralrimivva 2512 . . 3 (𝜑 → ∀𝑎R𝑏R (𝑎 <R 𝑏 → (∃𝑐𝐵 𝑎 <R 𝑐 ∨ ∀𝑐𝐵 𝑐 <R 𝑏)))
10416, 40, 103suplocsr 7610 . 2 (𝜑 → ∃𝑎R (∀𝑏𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)))
105 simprl 520 . . . 4 ((𝜑 ∧ (𝑎R ∧ (∀𝑏𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)))) → 𝑎R)
106105, 58sylibr 133 . . 3 ((𝜑 ∧ (𝑎R ∧ (∀𝑏𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)))) → ⟨𝑎, 0R⟩ ∈ ℝ)
107 breq2 3928 . . . . . . . 8 (𝑏 = (1st𝑦) → (𝑎 <R 𝑏𝑎 <R (1st𝑦)))
108107notbid 656 . . . . . . 7 (𝑏 = (1st𝑦) → (¬ 𝑎 <R 𝑏 ↔ ¬ 𝑎 <R (1st𝑦)))
109 simplrr 525 . . . . . . 7 (((𝜑 ∧ (𝑎R ∧ ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦𝐴) → ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)
1101sselda 3092 . . . . . . . . . . 11 ((𝜑𝑦𝐴) → 𝑦 ∈ ℝ)
111 elreal2 7631 . . . . . . . . . . 11 (𝑦 ∈ ℝ ↔ ((1st𝑦) ∈ R𝑦 = ⟨(1st𝑦), 0R⟩))
112110, 111sylib 121 . . . . . . . . . 10 ((𝜑𝑦𝐴) → ((1st𝑦) ∈ R𝑦 = ⟨(1st𝑦), 0R⟩))
113112simpld 111 . . . . . . . . 9 ((𝜑𝑦𝐴) → (1st𝑦) ∈ R)
114112simprd 113 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝑦 = ⟨(1st𝑦), 0R⟩)
115 simpr 109 . . . . . . . . . 10 ((𝜑𝑦𝐴) → 𝑦𝐴)
116114, 115eqeltrrd 2215 . . . . . . . . 9 ((𝜑𝑦𝐴) → ⟨(1st𝑦), 0R⟩ ∈ 𝐴)
117 opeq1 3700 . . . . . . . . . . 11 (𝑤 = (1st𝑦) → ⟨𝑤, 0R⟩ = ⟨(1st𝑦), 0R⟩)
118117eleq1d 2206 . . . . . . . . . 10 (𝑤 = (1st𝑦) → (⟨𝑤, 0R⟩ ∈ 𝐴 ↔ ⟨(1st𝑦), 0R⟩ ∈ 𝐴))
119118, 11elrab2 2838 . . . . . . . . 9 ((1st𝑦) ∈ 𝐵 ↔ ((1st𝑦) ∈ R ∧ ⟨(1st𝑦), 0R⟩ ∈ 𝐴))
120113, 116, 119sylanbrc 413 . . . . . . . 8 ((𝜑𝑦𝐴) → (1st𝑦) ∈ 𝐵)
121120adantlr 468 . . . . . . 7 (((𝜑 ∧ (𝑎R ∧ ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦𝐴) → (1st𝑦) ∈ 𝐵)
122108, 109, 121rspcdva 2789 . . . . . 6 (((𝜑 ∧ (𝑎R ∧ ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦𝐴) → ¬ 𝑎 <R (1st𝑦))
123114adantlr 468 . . . . . . . 8 (((𝜑 ∧ (𝑎R ∧ ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦𝐴) → 𝑦 = ⟨(1st𝑦), 0R⟩)
124123breq2d 3936 . . . . . . 7 (((𝜑 ∧ (𝑎R ∧ ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦𝐴) → (⟨𝑎, 0R⟩ < 𝑦 ↔ ⟨𝑎, 0R⟩ < ⟨(1st𝑦), 0R⟩))
125 ltresr 7640 . . . . . . 7 (⟨𝑎, 0R⟩ < ⟨(1st𝑦), 0R⟩ ↔ 𝑎 <R (1st𝑦))
126124, 125syl6bb 195 . . . . . 6 (((𝜑 ∧ (𝑎R ∧ ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦𝐴) → (⟨𝑎, 0R⟩ < 𝑦𝑎 <R (1st𝑦)))
127122, 126mtbird 662 . . . . 5 (((𝜑 ∧ (𝑎R ∧ ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)) ∧ 𝑦𝐴) → ¬ ⟨𝑎, 0R⟩ < 𝑦)
128127ralrimiva 2503 . . . 4 ((𝜑 ∧ (𝑎R ∧ ∀𝑏𝐵 ¬ 𝑎 <R 𝑏)) → ∀𝑦𝐴 ¬ ⟨𝑎, 0R⟩ < 𝑦)
129128adantrrr 478 . . 3 ((𝜑 ∧ (𝑎R ∧ (∀𝑏𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)))) → ∀𝑦𝐴 ¬ ⟨𝑎, 0R⟩ < 𝑦)
130 simplr 519 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → 𝑦 ∈ ℝ)
131130, 111sylib 121 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → ((1st𝑦) ∈ R𝑦 = ⟨(1st𝑦), 0R⟩))
132131simprd 113 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → 𝑦 = ⟨(1st𝑦), 0R⟩)
133 simpr 109 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → 𝑦 <𝑎, 0R⟩)
134132, 133eqbrtrrd 3947 . . . . . . . . 9 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → ⟨(1st𝑦), 0R⟩ <𝑎, 0R⟩)
135 ltresr 7640 . . . . . . . . 9 (⟨(1st𝑦), 0R⟩ <𝑎, 0R⟩ ↔ (1st𝑦) <R 𝑎)
136134, 135sylib 121 . . . . . . . 8 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → (1st𝑦) <R 𝑎)
137 breq1 3927 . . . . . . . . . 10 (𝑏 = (1st𝑦) → (𝑏 <R 𝑎 ↔ (1st𝑦) <R 𝑎))
138 breq1 3927 . . . . . . . . . . 11 (𝑏 = (1st𝑦) → (𝑏 <R 𝑐 ↔ (1st𝑦) <R 𝑐))
139138rexbidv 2436 . . . . . . . . . 10 (𝑏 = (1st𝑦) → (∃𝑐𝐵 𝑏 <R 𝑐 ↔ ∃𝑐𝐵 (1st𝑦) <R 𝑐))
140137, 139imbi12d 233 . . . . . . . . 9 (𝑏 = (1st𝑦) → ((𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐) ↔ ((1st𝑦) <R 𝑎 → ∃𝑐𝐵 (1st𝑦) <R 𝑐)))
141 simprr 521 . . . . . . . . . 10 ((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) → ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))
142141ad2antrr 479 . . . . . . . . 9 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))
143131simpld 111 . . . . . . . . 9 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → (1st𝑦) ∈ R)
144140, 142, 143rspcdva 2789 . . . . . . . 8 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → ((1st𝑦) <R 𝑎 → ∃𝑐𝐵 (1st𝑦) <R 𝑐))
145136, 144mpd 13 . . . . . . 7 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → ∃𝑐𝐵 (1st𝑦) <R 𝑐)
146 nfv 1508 . . . . . . . . . . 11 𝑐𝜑
147 nfv 1508 . . . . . . . . . . . 12 𝑐 𝑎R
148 nfcv 2279 . . . . . . . . . . . . 13 𝑐R
149 nfv 1508 . . . . . . . . . . . . . 14 𝑐 𝑏 <R 𝑎
150 nfre1 2474 . . . . . . . . . . . . . 14 𝑐𝑐𝐵 𝑏 <R 𝑐
151149, 150nfim 1551 . . . . . . . . . . . . 13 𝑐(𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)
152148, 151nfralya 2471 . . . . . . . . . . . 12 𝑐𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)
153147, 152nfan 1544 . . . . . . . . . . 11 𝑐(𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))
154146, 153nfan 1544 . . . . . . . . . 10 𝑐(𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)))
155 nfv 1508 . . . . . . . . . 10 𝑐 𝑦 ∈ ℝ
156154, 155nfan 1544 . . . . . . . . 9 𝑐((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ)
157 nfv 1508 . . . . . . . . 9 𝑐 𝑦 <𝑎, 0R
158156, 157nfan 1544 . . . . . . . 8 𝑐(((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩)
159 nfv 1508 . . . . . . . 8 𝑐𝑧𝐴 𝑦 < 𝑧
160 simprl 520 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) ∧ (𝑐𝐵 ∧ (1st𝑦) <R 𝑐)) → 𝑐𝐵)
161160, 92sylib 121 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) ∧ (𝑐𝐵 ∧ (1st𝑦) <R 𝑐)) → (𝑐R ∧ ⟨𝑐, 0R⟩ ∈ 𝐴))
162161simprd 113 . . . . . . . . . 10 (((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) ∧ (𝑐𝐵 ∧ (1st𝑦) <R 𝑐)) → ⟨𝑐, 0R⟩ ∈ 𝐴)
163132adantr 274 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) ∧ (𝑐𝐵 ∧ (1st𝑦) <R 𝑐)) → 𝑦 = ⟨(1st𝑦), 0R⟩)
164 simprr 521 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) ∧ (𝑐𝐵 ∧ (1st𝑦) <R 𝑐)) → (1st𝑦) <R 𝑐)
165 ltresr 7640 . . . . . . . . . . . 12 (⟨(1st𝑦), 0R⟩ <𝑐, 0R⟩ ↔ (1st𝑦) <R 𝑐)
166164, 165sylibr 133 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) ∧ (𝑐𝐵 ∧ (1st𝑦) <R 𝑐)) → ⟨(1st𝑦), 0R⟩ <𝑐, 0R⟩)
167163, 166eqbrtrd 3945 . . . . . . . . . 10 (((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) ∧ (𝑐𝐵 ∧ (1st𝑦) <R 𝑐)) → 𝑦 <𝑐, 0R⟩)
168 breq2 3928 . . . . . . . . . . 11 (𝑧 = ⟨𝑐, 0R⟩ → (𝑦 < 𝑧𝑦 <𝑐, 0R⟩))
169168rspcev 2784 . . . . . . . . . 10 ((⟨𝑐, 0R⟩ ∈ 𝐴𝑦 <𝑐, 0R⟩) → ∃𝑧𝐴 𝑦 < 𝑧)
170162, 167, 169syl2anc 408 . . . . . . . . 9 (((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) ∧ (𝑐𝐵 ∧ (1st𝑦) <R 𝑐)) → ∃𝑧𝐴 𝑦 < 𝑧)
171170exp32 362 . . . . . . . 8 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → (𝑐𝐵 → ((1st𝑦) <R 𝑐 → ∃𝑧𝐴 𝑦 < 𝑧)))
172158, 159, 171rexlimd 2544 . . . . . . 7 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → (∃𝑐𝐵 (1st𝑦) <R 𝑐 → ∃𝑧𝐴 𝑦 < 𝑧))
173145, 172mpd 13 . . . . . 6 ((((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 <𝑎, 0R⟩) → ∃𝑧𝐴 𝑦 < 𝑧)
174173ex 114 . . . . 5 (((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) ∧ 𝑦 ∈ ℝ) → (𝑦 <𝑎, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))
175174ralrimiva 2503 . . . 4 ((𝜑 ∧ (𝑎R ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐))) → ∀𝑦 ∈ ℝ (𝑦 <𝑎, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))
176175adantrrl 477 . . 3 ((𝜑 ∧ (𝑎R ∧ (∀𝑏𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)))) → ∀𝑦 ∈ ℝ (𝑦 <𝑎, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))
17749notbid 656 . . . . . 6 (𝑥 = ⟨𝑎, 0R⟩ → (¬ 𝑥 < 𝑦 ↔ ¬ ⟨𝑎, 0R⟩ < 𝑦))
178177ralbidv 2435 . . . . 5 (𝑥 = ⟨𝑎, 0R⟩ → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 ¬ ⟨𝑎, 0R⟩ < 𝑦))
179 breq2 3928 . . . . . . 7 (𝑥 = ⟨𝑎, 0R⟩ → (𝑦 < 𝑥𝑦 <𝑎, 0R⟩))
180179imbi1d 230 . . . . . 6 (𝑥 = ⟨𝑎, 0R⟩ → ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑦 <𝑎, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
181180ralbidv 2435 . . . . 5 (𝑥 = ⟨𝑎, 0R⟩ → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 <𝑎, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
182178, 181anbi12d 464 . . . 4 (𝑥 = ⟨𝑎, 0R⟩ → ((∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)) ↔ (∀𝑦𝐴 ¬ ⟨𝑎, 0R⟩ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <𝑎, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
183182rspcev 2784 . . 3 ((⟨𝑎, 0R⟩ ∈ ℝ ∧ (∀𝑦𝐴 ¬ ⟨𝑎, 0R⟩ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <𝑎, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
184106, 129, 176, 183syl12anc 1214 . 2 ((𝜑 ∧ (𝑎R ∧ (∀𝑏𝐵 ¬ 𝑎 <R 𝑏 ∧ ∀𝑏R (𝑏 <R 𝑎 → ∃𝑐𝐵 𝑏 <R 𝑐)))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
185104, 184rexlimddv 2552 1 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697   = wceq 1331  wex 1468  wcel 1480  wral 2414  wrex 2415  {crab 2418  wss 3066  cop 3525   class class class wbr 3924  cfv 5118  1st c1st 6029  Rcnr 7098  0Rc0r 7099   <R cltr 7104  cr 7612   < cltrr 7617
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-imp 7270  df-iltp 7271  df-enr 7527  df-nr 7528  df-plr 7529  df-mr 7530  df-ltr 7531  df-0r 7532  df-1r 7533  df-m1r 7534  df-r 7623  df-lt 7626
This theorem is referenced by:  axpre-suploc  7703
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