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Theorem axpre-sup 10579
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 10704. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10603. (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 10542 . . . . . . 7 (𝑥 ∈ ℝ ↔ ((1st𝑥) ∈ R𝑥 = ⟨(1st𝑥), 0R⟩))
21simplbi 498 . . . . . 6 (𝑥 ∈ ℝ → (1st𝑥) ∈ R)
32adantl 482 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (1st𝑥) ∈ R)
4 fo1st 7698 . . . . . . . . . . . 12 1st :V–onto→V
5 fof 6583 . . . . . . . . . . . 12 (1st :V–onto→V → 1st :V⟶V)
6 ffn 6507 . . . . . . . . . . . 12 (1st :V⟶V → 1st Fn V)
74, 5, 6mp2b 10 . . . . . . . . . . 11 1st Fn V
8 ssv 3988 . . . . . . . . . . 11 𝐴 ⊆ V
9 fvelimab 6730 . . . . . . . . . . 11 ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑤 ∈ (1st𝐴) ↔ ∃𝑦𝐴 (1st𝑦) = 𝑤))
107, 8, 9mp2an 688 . . . . . . . . . 10 (𝑤 ∈ (1st𝐴) ↔ ∃𝑦𝐴 (1st𝑦) = 𝑤)
11 r19.29 3251 . . . . . . . . . . . 12 ((∀𝑦𝐴 𝑦 < 𝑥 ∧ ∃𝑦𝐴 (1st𝑦) = 𝑤) → ∃𝑦𝐴 (𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤))
12 ssel2 3959 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → 𝑦 ∈ ℝ)
13 ltresr2 10551 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ (1st𝑦) <R (1st𝑥)))
14 breq1 5060 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑦) = 𝑤 → ((1st𝑦) <R (1st𝑥) ↔ 𝑤 <R (1st𝑥)))
1513, 14sylan9bb 510 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st𝑦) = 𝑤) → (𝑦 < 𝑥𝑤 <R (1st𝑥)))
1615biimpd 230 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st𝑦) = 𝑤) → (𝑦 < 𝑥𝑤 <R (1st𝑥)))
1716exp31 420 . . . . . . . . . . . . . . . . 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 422 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → (((1st𝑦) = 𝑤𝑦 < 𝑥) → 𝑤 <R (1st𝑥)))
2019ancomsd 466 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → ((𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤) → 𝑤 <R (1st𝑥)))
2120an32s 648 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝐴) → ((𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤) → 𝑤 <R (1st𝑥)))
2221rexlimdva 3281 . . . . . . . . . . . 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 416 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → (∃𝑦𝐴 (1st𝑦) = 𝑤𝑤 <R (1st𝑥))))
2510, 24syl7bi 256 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥))))
2625impr 455 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥)))
2726adantlr 711 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥)))
2827ralrimiv 3178 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥))
2928expr 457 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥)))
30 brralrspcev 5117 . . . . 5 (((1st𝑥) ∈ R ∧ ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥)) → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣)
313, 29, 30syl6an 680 . . . 4 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣))
3231rexlimdva 3281 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣))
33 n0 4307 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
34 fnfvima 6986 . . . . . . . . 9 ((1st Fn V ∧ 𝐴 ⊆ V ∧ 𝑦𝐴) → (1st𝑦) ∈ (1st𝐴))
357, 8, 34mp3an12 1442 . . . . . . . 8 (𝑦𝐴 → (1st𝑦) ∈ (1st𝐴))
3635ne0d 4298 . . . . . . 7 (𝑦𝐴 → (1st𝐴) ≠ ∅)
3736exlimiv 1922 . . . . . 6 (∃𝑦 𝑦𝐴 → (1st𝐴) ≠ ∅)
3833, 37sylbi 218 . . . . 5 (𝐴 ≠ ∅ → (1st𝐴) ≠ ∅)
39 supsr 10522 . . . . . 6 (((1st𝐴) ≠ ∅ ∧ ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣) → ∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢)))
4039ex 413 . . . . 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 482 . . 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 5061 . . . . . . . . . . . 12 (𝑤 = (1st𝑦) → (𝑣 <R 𝑤𝑣 <R (1st𝑦)))
4443notbid 319 . . . . . . . . . . 11 (𝑤 = (1st𝑦) → (¬ 𝑣 <R 𝑤 ↔ ¬ 𝑣 <R (1st𝑦)))
4544rspccv 3617 . . . . . . . . . 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 482 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st𝑦)))
48 elreal2 10542 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ ↔ ((1st𝑦) ∈ R𝑦 = ⟨(1st𝑦), 0R⟩))
4948simprbi 497 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 = ⟨(1st𝑦), 0R⟩)
5049breq2d 5069 . . . . . . . . . . 11 (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ < 𝑦 ↔ ⟨𝑣, 0R⟩ < ⟨(1st𝑦), 0R⟩))
51 ltresr 10550 . . . . . . . . . . 11 (⟨𝑣, 0R⟩ < ⟨(1st𝑦), 0R⟩ ↔ 𝑣 <R (1st𝑦))
5250, 51syl6bb 288 . . . . . . . . . 10 (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ < 𝑦𝑣 <R (1st𝑦)))
5312, 52syl 17 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (⟨𝑣, 0R⟩ < 𝑦𝑣 <R (1st𝑦)))
5453notbid 319 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (¬ ⟨𝑣, 0R⟩ < 𝑦 ↔ ¬ 𝑣 <R (1st𝑦)))
5547, 54sylibrd 260 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ¬ ⟨𝑣, 0R⟩ < 𝑦))
5655ralrimdva 3186 . . . . . 6 (𝐴 ⊆ ℝ → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
5756ad2antrr 722 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
5849breq1d 5067 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ ↔ ⟨(1st𝑦), 0R⟩ <𝑣, 0R⟩))
59 ltresr 10550 . . . . . . . . . . . . . 14 (⟨(1st𝑦), 0R⟩ <𝑣, 0R⟩ ↔ (1st𝑦) <R 𝑣)
6058, 59syl6bb 288 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ ↔ (1st𝑦) <R 𝑣))
6148simplbi 498 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → (1st𝑦) ∈ R)
62 breq1 5060 . . . . . . . . . . . . . . . . 17 (𝑤 = (1st𝑦) → (𝑤 <R 𝑣 ↔ (1st𝑦) <R 𝑣))
63 breq1 5060 . . . . . . . . . . . . . . . . . 18 (𝑤 = (1st𝑦) → (𝑤 <R 𝑢 ↔ (1st𝑦) <R 𝑢))
6463rexbidv 3294 . . . . . . . . . . . . . . . . 17 (𝑤 = (1st𝑦) → (∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢 ↔ ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢))
6562, 64imbi12d 346 . . . . . . . . . . . . . . . 16 (𝑤 = (1st𝑦) → ((𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) ↔ ((1st𝑦) <R 𝑣 → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
6665rspccv 3617 . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
7069adantr 481 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑦 <𝑣, 0R⟩ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
71 fvelimab 6730 . . . . . . . . . . . . . . . 16 ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑢 ∈ (1st𝐴) ↔ ∃𝑧𝐴 (1st𝑧) = 𝑢))
727, 8, 71mp2an 688 . . . . . . . . . . . . . . 15 (𝑢 ∈ (1st𝐴) ↔ ∃𝑧𝐴 (1st𝑧) = 𝑢)
73 ssel2 3959 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ 𝑧𝐴) → 𝑧 ∈ ℝ)
74 ltresr2 10551 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 < 𝑧 ↔ (1st𝑦) <R (1st𝑧)))
7573, 74sylan2 592 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) → (𝑦 < 𝑧 ↔ (1st𝑦) <R (1st𝑧)))
76 breq2 5061 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑧) = 𝑢 → ((1st𝑦) <R (1st𝑧) ↔ (1st𝑦) <R 𝑢))
7775, 76sylan9bb 510 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) ∧ (1st𝑧) = 𝑢) → (𝑦 < 𝑧 ↔ (1st𝑦) <R 𝑢))
7877exbiri 807 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) → ((1st𝑧) = 𝑢 → ((1st𝑦) <R 𝑢𝑦 < 𝑧)))
7978expr 457 . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . 16 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑧𝐴 → ((1st𝑧) = 𝑢𝑦 < 𝑧)))
8281reximdvai 3269 . . . . . . . . . . . . . . 15 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (∃𝑧𝐴 (1st𝑧) = 𝑢 → ∃𝑧𝐴 𝑦 < 𝑧))
8372, 82syl5bi 243 . . . . . . . . . . . . . 14 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑢 ∈ (1st𝐴) → ∃𝑧𝐴 𝑦 < 𝑧))
8483expcom 414 . . . . . . . . . . . . 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 3280 . . . . . . . . . . 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 413 . . . . . . . 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 722 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
9291ralrimdv 3185 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
93 opelreal 10540 . . . . . . . 8 (⟨𝑣, 0R⟩ ∈ ℝ ↔ 𝑣R)
9493biimpri 229 . . . . . . 7 (𝑣R → ⟨𝑣, 0R⟩ ∈ ℝ)
9594adantl 482 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → ⟨𝑣, 0R⟩ ∈ ℝ)
96 breq1 5060 . . . . . . . . . . 11 (𝑥 = ⟨𝑣, 0R⟩ → (𝑥 < 𝑦 ↔ ⟨𝑣, 0R⟩ < 𝑦))
9796notbid 319 . . . . . . . . . 10 (𝑥 = ⟨𝑣, 0R⟩ → (¬ 𝑥 < 𝑦 ↔ ¬ ⟨𝑣, 0R⟩ < 𝑦))
9897ralbidv 3194 . . . . . . . . 9 (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
99 breq2 5061 . . . . . . . . . . 11 (𝑥 = ⟨𝑣, 0R⟩ → (𝑦 < 𝑥𝑦 <𝑣, 0R⟩))
10099imbi1d 343 . . . . . . . . . 10 (𝑥 = ⟨𝑣, 0R⟩ → ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
101100ralbidv 3194 . . . . . . . . 9 (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
10298, 101anbi12d 630 . . . . . . . 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 3620 . . . . . . 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 413 . . . . . 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 607 . . . 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 3281 . . 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 1109 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 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  wss 3933  c0 4288  cop 4563   class class class wbr 5057  cima 5551   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  1st c1st 7676  Rcnr 10275  0Rc0r 10276   <R cltr 10281  cr 10524   < cltrr 10529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-er 8278  df-ec 8280  df-qs 8284  df-ni 10282  df-pli 10283  df-mi 10284  df-lti 10285  df-plpq 10318  df-mpq 10319  df-ltpq 10320  df-enq 10321  df-nq 10322  df-erq 10323  df-plq 10324  df-mq 10325  df-1nq 10326  df-rq 10327  df-ltnq 10328  df-np 10391  df-1p 10392  df-plp 10393  df-mp 10394  df-ltp 10395  df-enr 10465  df-nr 10466  df-plr 10467  df-mr 10468  df-ltr 10469  df-0r 10470  df-1r 10471  df-m1r 10472  df-r 10535  df-lt 10538
This theorem is referenced by: (None)
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