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Theorem axpre-sup 11206
Description: A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 11333. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11230. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-sup ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem axpre-sup
Dummy variables 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal2 11169 . . . . . . 7 (𝑥 ∈ ℝ ↔ ((1st𝑥) ∈ R𝑥 = ⟨(1st𝑥), 0R⟩))
21simplbi 497 . . . . . 6 (𝑥 ∈ ℝ → (1st𝑥) ∈ R)
32adantl 481 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (1st𝑥) ∈ R)
4 fo1st 8032 . . . . . . . . . . . 12 1st :V–onto→V
5 fof 6820 . . . . . . . . . . . 12 (1st :V–onto→V → 1st :V⟶V)
6 ffn 6736 . . . . . . . . . . . 12 (1st :V⟶V → 1st Fn V)
74, 5, 6mp2b 10 . . . . . . . . . . 11 1st Fn V
8 ssv 4019 . . . . . . . . . . 11 𝐴 ⊆ V
9 fvelimab 6980 . . . . . . . . . . 11 ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑤 ∈ (1st𝐴) ↔ ∃𝑦𝐴 (1st𝑦) = 𝑤))
107, 8, 9mp2an 692 . . . . . . . . . 10 (𝑤 ∈ (1st𝐴) ↔ ∃𝑦𝐴 (1st𝑦) = 𝑤)
11 r19.29 3111 . . . . . . . . . . . 12 ((∀𝑦𝐴 𝑦 < 𝑥 ∧ ∃𝑦𝐴 (1st𝑦) = 𝑤) → ∃𝑦𝐴 (𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤))
12 ssel2 3989 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → 𝑦 ∈ ℝ)
13 ltresr2 11178 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ (1st𝑦) <R (1st𝑥)))
14 breq1 5150 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑦) = 𝑤 → ((1st𝑦) <R (1st𝑥) ↔ 𝑤 <R (1st𝑥)))
1513, 14sylan9bb 509 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st𝑦) = 𝑤) → (𝑦 < 𝑥𝑤 <R (1st𝑥)))
1615biimpd 229 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st𝑦) = 𝑤) → (𝑦 < 𝑥𝑤 <R (1st𝑥)))
1716exp31 419 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ → (𝑥 ∈ ℝ → ((1st𝑦) = 𝑤 → (𝑦 < 𝑥𝑤 <R (1st𝑥)))))
1812, 17syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (𝑥 ∈ ℝ → ((1st𝑦) = 𝑤 → (𝑦 < 𝑥𝑤 <R (1st𝑥)))))
1918imp4b 421 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → (((1st𝑦) = 𝑤𝑦 < 𝑥) → 𝑤 <R (1st𝑥)))
2019ancomsd 465 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → ((𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤) → 𝑤 <R (1st𝑥)))
2120an32s 652 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝐴) → ((𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤) → 𝑤 <R (1st𝑥)))
2221rexlimdva 3152 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∃𝑦𝐴 (𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤) → 𝑤 <R (1st𝑥)))
2311, 22syl5 34 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → ((∀𝑦𝐴 𝑦 < 𝑥 ∧ ∃𝑦𝐴 (1st𝑦) = 𝑤) → 𝑤 <R (1st𝑥)))
2423expd 415 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → (∃𝑦𝐴 (1st𝑦) = 𝑤𝑤 <R (1st𝑥))))
2510, 24syl7bi 255 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥))))
2625impr 454 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥)))
2726adantlr 715 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥)))
2827ralrimiv 3142 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥))
2928expr 456 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥)))
30 brralrspcev 5207 . . . . 5 (((1st𝑥) ∈ R ∧ ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥)) → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣)
313, 29, 30syl6an 684 . . . 4 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣))
3231rexlimdva 3152 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣))
33 n0 4358 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
34 fnfvima 7252 . . . . . . . . 9 ((1st Fn V ∧ 𝐴 ⊆ V ∧ 𝑦𝐴) → (1st𝑦) ∈ (1st𝐴))
357, 8, 34mp3an12 1450 . . . . . . . 8 (𝑦𝐴 → (1st𝑦) ∈ (1st𝐴))
3635ne0d 4347 . . . . . . 7 (𝑦𝐴 → (1st𝐴) ≠ ∅)
3736exlimiv 1927 . . . . . 6 (∃𝑦 𝑦𝐴 → (1st𝐴) ≠ ∅)
3833, 37sylbi 217 . . . . 5 (𝐴 ≠ ∅ → (1st𝐴) ≠ ∅)
39 supsr 11149 . . . . . 6 (((1st𝐴) ≠ ∅ ∧ ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣) → ∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢)))
4039ex 412 . . . . 5 ((1st𝐴) ≠ ∅ → (∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣 → ∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢))))
4138, 40syl 17 . . . 4 (𝐴 ≠ ∅ → (∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣 → ∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢))))
4241adantl 481 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣 → ∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢))))
43 breq2 5151 . . . . . . . . . . . 12 (𝑤 = (1st𝑦) → (𝑣 <R 𝑤𝑣 <R (1st𝑦)))
4443notbid 318 . . . . . . . . . . 11 (𝑤 = (1st𝑦) → (¬ 𝑣 <R 𝑤 ↔ ¬ 𝑣 <R (1st𝑦)))
4544rspccv 3618 . . . . . . . . . 10 (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ((1st𝑦) ∈ (1st𝐴) → ¬ 𝑣 <R (1st𝑦)))
4635, 45syl5com 31 . . . . . . . . 9 (𝑦𝐴 → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st𝑦)))
4746adantl 481 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st𝑦)))
48 elreal2 11169 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ ↔ ((1st𝑦) ∈ R𝑦 = ⟨(1st𝑦), 0R⟩))
4948simprbi 496 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 = ⟨(1st𝑦), 0R⟩)
5049breq2d 5159 . . . . . . . . . . 11 (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ < 𝑦 ↔ ⟨𝑣, 0R⟩ < ⟨(1st𝑦), 0R⟩))
51 ltresr 11177 . . . . . . . . . . 11 (⟨𝑣, 0R⟩ < ⟨(1st𝑦), 0R⟩ ↔ 𝑣 <R (1st𝑦))
5250, 51bitrdi 287 . . . . . . . . . 10 (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ < 𝑦𝑣 <R (1st𝑦)))
5312, 52syl 17 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (⟨𝑣, 0R⟩ < 𝑦𝑣 <R (1st𝑦)))
5453notbid 318 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (¬ ⟨𝑣, 0R⟩ < 𝑦 ↔ ¬ 𝑣 <R (1st𝑦)))
5547, 54sylibrd 259 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ¬ ⟨𝑣, 0R⟩ < 𝑦))
5655ralrimdva 3151 . . . . . 6 (𝐴 ⊆ ℝ → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
5756ad2antrr 726 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
5849breq1d 5157 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ ↔ ⟨(1st𝑦), 0R⟩ <𝑣, 0R⟩))
59 ltresr 11177 . . . . . . . . . . . . . 14 (⟨(1st𝑦), 0R⟩ <𝑣, 0R⟩ ↔ (1st𝑦) <R 𝑣)
6058, 59bitrdi 287 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ ↔ (1st𝑦) <R 𝑣))
6148simplbi 497 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → (1st𝑦) ∈ R)
62 breq1 5150 . . . . . . . . . . . . . . . . 17 (𝑤 = (1st𝑦) → (𝑤 <R 𝑣 ↔ (1st𝑦) <R 𝑣))
63 breq1 5150 . . . . . . . . . . . . . . . . . 18 (𝑤 = (1st𝑦) → (𝑤 <R 𝑢 ↔ (1st𝑦) <R 𝑢))
6463rexbidv 3176 . . . . . . . . . . . . . . . . 17 (𝑤 = (1st𝑦) → (∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢 ↔ ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢))
6562, 64imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑤 = (1st𝑦) → ((𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) ↔ ((1st𝑦) <R 𝑣 → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
6665rspccv 3618 . . . . . . . . . . . . . . 15 (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ((1st𝑦) ∈ R → ((1st𝑦) <R 𝑣 → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
6761, 66syl5 34 . . . . . . . . . . . . . 14 (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → ((1st𝑦) <R 𝑣 → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
6867com3l 89 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → ((1st𝑦) <R 𝑣 → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
6960, 68sylbid 240 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
7069adantr 480 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑦 <𝑣, 0R⟩ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
71 fvelimab 6980 . . . . . . . . . . . . . . . 16 ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑢 ∈ (1st𝐴) ↔ ∃𝑧𝐴 (1st𝑧) = 𝑢))
727, 8, 71mp2an 692 . . . . . . . . . . . . . . 15 (𝑢 ∈ (1st𝐴) ↔ ∃𝑧𝐴 (1st𝑧) = 𝑢)
73 ssel2 3989 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ 𝑧𝐴) → 𝑧 ∈ ℝ)
74 ltresr2 11178 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 < 𝑧 ↔ (1st𝑦) <R (1st𝑧)))
7573, 74sylan2 593 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) → (𝑦 < 𝑧 ↔ (1st𝑦) <R (1st𝑧)))
76 breq2 5151 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑧) = 𝑢 → ((1st𝑦) <R (1st𝑧) ↔ (1st𝑦) <R 𝑢))
7775, 76sylan9bb 509 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) ∧ (1st𝑧) = 𝑢) → (𝑦 < 𝑧 ↔ (1st𝑦) <R 𝑢))
7877exbiri 811 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) → ((1st𝑧) = 𝑢 → ((1st𝑦) <R 𝑢𝑦 < 𝑧)))
7978expr 456 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧𝐴 → ((1st𝑧) = 𝑢 → ((1st𝑦) <R 𝑢𝑦 < 𝑧))))
8079com4r 94 . . . . . . . . . . . . . . . . 17 ((1st𝑦) <R 𝑢 → ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧𝐴 → ((1st𝑧) = 𝑢𝑦 < 𝑧))))
8180imp 406 . . . . . . . . . . . . . . . 16 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑧𝐴 → ((1st𝑧) = 𝑢𝑦 < 𝑧)))
8281reximdvai 3162 . . . . . . . . . . . . . . 15 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (∃𝑧𝐴 (1st𝑧) = 𝑢 → ∃𝑧𝐴 𝑦 < 𝑧))
8372, 82biimtrid 242 . . . . . . . . . . . . . 14 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑢 ∈ (1st𝐴) → ∃𝑧𝐴 𝑦 < 𝑧))
8483expcom 413 . . . . . . . . . . . . 13 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → ((1st𝑦) <R 𝑢 → (𝑢 ∈ (1st𝐴) → ∃𝑧𝐴 𝑦 < 𝑧)))
8584com23 86 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑢 ∈ (1st𝐴) → ((1st𝑦) <R 𝑢 → ∃𝑧𝐴 𝑦 < 𝑧)))
8685rexlimdv 3150 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢 → ∃𝑧𝐴 𝑦 < 𝑧))
8770, 86syl6d 75 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑦 <𝑣, 0R⟩ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑧𝐴 𝑦 < 𝑧)))
8887com23 86 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
8988ex 412 . . . . . . . 8 (𝑦 ∈ ℝ → (𝐴 ⊆ ℝ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
9089com3l 89 . . . . . . 7 (𝐴 ⊆ ℝ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
9190ad2antrr 726 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
9291ralrimdv 3149 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
93 opelreal 11167 . . . . . . . 8 (⟨𝑣, 0R⟩ ∈ ℝ ↔ 𝑣R)
9493biimpri 228 . . . . . . 7 (𝑣R → ⟨𝑣, 0R⟩ ∈ ℝ)
9594adantl 481 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → ⟨𝑣, 0R⟩ ∈ ℝ)
96 breq1 5150 . . . . . . . . . . 11 (𝑥 = ⟨𝑣, 0R⟩ → (𝑥 < 𝑦 ↔ ⟨𝑣, 0R⟩ < 𝑦))
9796notbid 318 . . . . . . . . . 10 (𝑥 = ⟨𝑣, 0R⟩ → (¬ 𝑥 < 𝑦 ↔ ¬ ⟨𝑣, 0R⟩ < 𝑦))
9897ralbidv 3175 . . . . . . . . 9 (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
99 breq2 5151 . . . . . . . . . . 11 (𝑥 = ⟨𝑣, 0R⟩ → (𝑦 < 𝑥𝑦 <𝑣, 0R⟩))
10099imbi1d 341 . . . . . . . . . 10 (𝑥 = ⟨𝑣, 0R⟩ → ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
101100ralbidv 3175 . . . . . . . . 9 (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
10298, 101anbi12d 632 . . . . . . . 8 (𝑥 = ⟨𝑣, 0R⟩ → ((∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)) ↔ (∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
103102rspcev 3621 . . . . . . 7 ((⟨𝑣, 0R⟩ ∈ ℝ ∧ (∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
104103ex 412 . . . . . 6 (⟨𝑣, 0R⟩ ∈ ℝ → ((∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))))
10595, 104syl 17 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → ((∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))))
10657, 92, 105syl2and 608 . . . 4 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → ((∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢)) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))))
107106rexlimdva 3152 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢)) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))))
10832, 42, 1073syld 60 . 2 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))))
1091083impia 1116 1 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  Vcvv 3477  wss 3962  c0 4338  cop 4636   class class class wbr 5147  cima 5691   Fn wfn 6557  wf 6558  ontowfo 6560  cfv 6562  1st c1st 8010  Rcnr 10902  0Rc0r 10903   <R cltr 10908  cr 11151   < cltrr 11156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-omul 8509  df-er 8743  df-ec 8745  df-qs 8749  df-ni 10909  df-pli 10910  df-mi 10911  df-lti 10912  df-plpq 10945  df-mpq 10946  df-ltpq 10947  df-enq 10948  df-nq 10949  df-erq 10950  df-plq 10951  df-mq 10952  df-1nq 10953  df-rq 10954  df-ltnq 10955  df-np 11018  df-1p 11019  df-plp 11020  df-mp 11021  df-ltp 11022  df-enr 11092  df-nr 11093  df-plr 11094  df-mr 11095  df-ltr 11096  df-0r 11097  df-1r 11098  df-m1r 11099  df-r 11162  df-lt 11165
This theorem is referenced by: (None)
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