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Theorem axpre-sup 10028
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 10151. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10052. (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 9991 . . . . . . 7 (𝑥 ∈ ℝ ↔ ((1st𝑥) ∈ R𝑥 = ⟨(1st𝑥), 0R⟩))
21simplbi 475 . . . . . 6 (𝑥 ∈ ℝ → (1st𝑥) ∈ R)
32adantl 481 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (1st𝑥) ∈ R)
4 fo1st 7230 . . . . . . . . . . . 12 1st :V–onto→V
5 fof 6153 . . . . . . . . . . . 12 (1st :V–onto→V → 1st :V⟶V)
6 ffn 6083 . . . . . . . . . . . 12 (1st :V⟶V → 1st Fn V)
74, 5, 6mp2b 10 . . . . . . . . . . 11 1st Fn V
8 ssv 3658 . . . . . . . . . . 11 𝐴 ⊆ V
9 fvelimab 6292 . . . . . . . . . . 11 ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑤 ∈ (1st𝐴) ↔ ∃𝑦𝐴 (1st𝑦) = 𝑤))
107, 8, 9mp2an 708 . . . . . . . . . 10 (𝑤 ∈ (1st𝐴) ↔ ∃𝑦𝐴 (1st𝑦) = 𝑤)
11 r19.29 3101 . . . . . . . . . . . 12 ((∀𝑦𝐴 𝑦 < 𝑥 ∧ ∃𝑦𝐴 (1st𝑦) = 𝑤) → ∃𝑦𝐴 (𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤))
12 ssel2 3631 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → 𝑦 ∈ ℝ)
13 ltresr2 10000 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ (1st𝑦) <R (1st𝑥)))
14 breq1 4688 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑦) = 𝑤 → ((1st𝑦) <R (1st𝑥) ↔ 𝑤 <R (1st𝑥)))
1513, 14sylan9bb 736 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st𝑦) = 𝑤) → (𝑦 < 𝑥𝑤 <R (1st𝑥)))
1615biimpd 219 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st𝑦) = 𝑤) → (𝑦 < 𝑥𝑤 <R (1st𝑥)))
1716exp31 629 . . . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → (((1st𝑦) = 𝑤𝑦 < 𝑥) → 𝑤 <R (1st𝑥)))
2019ancomsd 469 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → ((𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤) → 𝑤 <R (1st𝑥)))
2120an32s 863 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝐴) → ((𝑦 < 𝑥 ∧ (1st𝑦) = 𝑤) → 𝑤 <R (1st𝑥)))
2221rexlimdva 3060 . . . . . . . . . . . 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 451 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → (∃𝑦𝐴 (1st𝑦) = 𝑤𝑤 <R (1st𝑥))))
2510, 24syl7bi 245 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥))))
2625impr 648 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥)))
2726adantlr 751 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → (𝑤 ∈ (1st𝐴) → 𝑤 <R (1st𝑥)))
2827ralrimiv 2994 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦𝐴 𝑦 < 𝑥)) → ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥))
2928expr 642 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥)))
30 breq2 4689 . . . . . . 7 (𝑣 = (1st𝑥) → (𝑤 <R 𝑣𝑤 <R (1st𝑥)))
3130ralbidv 3015 . . . . . 6 (𝑣 = (1st𝑥) → (∀𝑤 ∈ (1st𝐴)𝑤 <R 𝑣 ↔ ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥)))
3231rspcev 3340 . . . . 5 (((1st𝑥) ∈ R ∧ ∀𝑤 ∈ (1st𝐴)𝑤 <R (1st𝑥)) → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣)
333, 29, 32syl6an 567 . . . 4 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 𝑦 < 𝑥 → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣))
3433rexlimdva 3060 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 → ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣))
35 n0 3964 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
36 fnfvima 6536 . . . . . . . . 9 ((1st Fn V ∧ 𝐴 ⊆ V ∧ 𝑦𝐴) → (1st𝑦) ∈ (1st𝐴))
377, 8, 36mp3an12 1454 . . . . . . . 8 (𝑦𝐴 → (1st𝑦) ∈ (1st𝐴))
38 ne0i 3954 . . . . . . . 8 ((1st𝑦) ∈ (1st𝐴) → (1st𝐴) ≠ ∅)
3937, 38syl 17 . . . . . . 7 (𝑦𝐴 → (1st𝐴) ≠ ∅)
4039exlimiv 1898 . . . . . 6 (∃𝑦 𝑦𝐴 → (1st𝐴) ≠ ∅)
4135, 40sylbi 207 . . . . 5 (𝐴 ≠ ∅ → (1st𝐴) ≠ ∅)
42 supsr 9971 . . . . . 6 (((1st𝐴) ≠ ∅ ∧ ∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣) → ∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢)))
4342ex 449 . . . . 5 ((1st𝐴) ≠ ∅ → (∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣 → ∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢))))
4441, 43syl 17 . . . 4 (𝐴 ≠ ∅ → (∃𝑣R𝑤 ∈ (1st𝐴)𝑤 <R 𝑣 → ∃𝑣R (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢))))
4544adantl 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))))
46 breq2 4689 . . . . . . . . . . . 12 (𝑤 = (1st𝑦) → (𝑣 <R 𝑤𝑣 <R (1st𝑦)))
4746notbid 307 . . . . . . . . . . 11 (𝑤 = (1st𝑦) → (¬ 𝑣 <R 𝑤 ↔ ¬ 𝑣 <R (1st𝑦)))
4847rspccv 3337 . . . . . . . . . 10 (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ((1st𝑦) ∈ (1st𝐴) → ¬ 𝑣 <R (1st𝑦)))
4937, 48syl5com 31 . . . . . . . . 9 (𝑦𝐴 → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st𝑦)))
5049adantl 481 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st𝑦)))
51 elreal2 9991 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ ↔ ((1st𝑦) ∈ R𝑦 = ⟨(1st𝑦), 0R⟩))
5251simprbi 479 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 = ⟨(1st𝑦), 0R⟩)
5352breq2d 4697 . . . . . . . . . . 11 (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ < 𝑦 ↔ ⟨𝑣, 0R⟩ < ⟨(1st𝑦), 0R⟩))
54 ltresr 9999 . . . . . . . . . . 11 (⟨𝑣, 0R⟩ < ⟨(1st𝑦), 0R⟩ ↔ 𝑣 <R (1st𝑦))
5553, 54syl6bb 276 . . . . . . . . . 10 (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ < 𝑦𝑣 <R (1st𝑦)))
5612, 55syl 17 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (⟨𝑣, 0R⟩ < 𝑦𝑣 <R (1st𝑦)))
5756notbid 307 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (¬ ⟨𝑣, 0R⟩ < 𝑦 ↔ ¬ 𝑣 <R (1st𝑦)))
5850, 57sylibrd 249 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ¬ ⟨𝑣, 0R⟩ < 𝑦))
5958ralrimdva 2998 . . . . . 6 (𝐴 ⊆ ℝ → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
6059ad2antrr 762 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤 ∈ (1st𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
6152breq1d 4695 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ ↔ ⟨(1st𝑦), 0R⟩ <𝑣, 0R⟩))
62 ltresr 9999 . . . . . . . . . . . . . 14 (⟨(1st𝑦), 0R⟩ <𝑣, 0R⟩ ↔ (1st𝑦) <R 𝑣)
6361, 62syl6bb 276 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ ↔ (1st𝑦) <R 𝑣))
6451simplbi 475 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → (1st𝑦) ∈ R)
65 breq1 4688 . . . . . . . . . . . . . . . . 17 (𝑤 = (1st𝑦) → (𝑤 <R 𝑣 ↔ (1st𝑦) <R 𝑣))
66 breq1 4688 . . . . . . . . . . . . . . . . . 18 (𝑤 = (1st𝑦) → (𝑤 <R 𝑢 ↔ (1st𝑦) <R 𝑢))
6766rexbidv 3081 . . . . . . . . . . . . . . . . 17 (𝑤 = (1st𝑦) → (∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢 ↔ ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢))
6865, 67imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑤 = (1st𝑦) → ((𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) ↔ ((1st𝑦) <R 𝑣 → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
6968rspccv 3337 . . . . . . . . . . . . . . 15 (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ((1st𝑦) ∈ R → ((1st𝑦) <R 𝑣 → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
7064, 69syl5 34 . . . . . . . . . . . . . 14 (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → ((1st𝑦) <R 𝑣 → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
7170com3l 89 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → ((1st𝑦) <R 𝑣 → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
7263, 71sylbid 230 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
7372adantr 480 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑦 <𝑣, 0R⟩ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢)))
74 fvelimab 6292 . . . . . . . . . . . . . . . 16 ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑢 ∈ (1st𝐴) ↔ ∃𝑧𝐴 (1st𝑧) = 𝑢))
757, 8, 74mp2an 708 . . . . . . . . . . . . . . 15 (𝑢 ∈ (1st𝐴) ↔ ∃𝑧𝐴 (1st𝑧) = 𝑢)
76 ssel2 3631 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ 𝑧𝐴) → 𝑧 ∈ ℝ)
77 ltresr2 10000 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 < 𝑧 ↔ (1st𝑦) <R (1st𝑧)))
7876, 77sylan2 490 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) → (𝑦 < 𝑧 ↔ (1st𝑦) <R (1st𝑧)))
79 breq2 4689 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑧) = 𝑢 → ((1st𝑦) <R (1st𝑧) ↔ (1st𝑦) <R 𝑢))
8078, 79sylan9bb 736 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) ∧ (1st𝑧) = 𝑢) → (𝑦 < 𝑧 ↔ (1st𝑦) <R 𝑢))
8180exbiri 651 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧𝐴)) → ((1st𝑧) = 𝑢 → ((1st𝑦) <R 𝑢𝑦 < 𝑧)))
8281expr 642 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧𝐴 → ((1st𝑧) = 𝑢 → ((1st𝑦) <R 𝑢𝑦 < 𝑧))))
8382com4r 94 . . . . . . . . . . . . . . . . 17 ((1st𝑦) <R 𝑢 → ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧𝐴 → ((1st𝑧) = 𝑢𝑦 < 𝑧))))
8483imp 444 . . . . . . . . . . . . . . . 16 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑧𝐴 → ((1st𝑧) = 𝑢𝑦 < 𝑧)))
8584reximdvai 3044 . . . . . . . . . . . . . . 15 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (∃𝑧𝐴 (1st𝑧) = 𝑢 → ∃𝑧𝐴 𝑦 < 𝑧))
8675, 85syl5bi 232 . . . . . . . . . . . . . 14 (((1st𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑢 ∈ (1st𝐴) → ∃𝑧𝐴 𝑦 < 𝑧))
8786expcom 450 . . . . . . . . . . . . 13 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → ((1st𝑦) <R 𝑢 → (𝑢 ∈ (1st𝐴) → ∃𝑧𝐴 𝑦 < 𝑧)))
8887com23 86 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑢 ∈ (1st𝐴) → ((1st𝑦) <R 𝑢 → ∃𝑧𝐴 𝑦 < 𝑧)))
8988rexlimdv 3059 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (∃𝑢 ∈ (1st𝐴)(1st𝑦) <R 𝑢 → ∃𝑧𝐴 𝑦 < 𝑧))
9073, 89syl6d 75 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑦 <𝑣, 0R⟩ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∃𝑧𝐴 𝑦 < 𝑧)))
9190com23 86 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
9291ex 449 . . . . . . . 8 (𝑦 ∈ ℝ → (𝐴 ⊆ ℝ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
9392com3l 89 . . . . . . 7 (𝐴 ⊆ ℝ → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
9493ad2antrr 762 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧))))
9594ralrimdv 2997 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → (∀𝑤R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st𝐴)𝑤 <R 𝑢) → ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
96 opelreal 9989 . . . . . . . 8 (⟨𝑣, 0R⟩ ∈ ℝ ↔ 𝑣R)
9796biimpri 218 . . . . . . 7 (𝑣R → ⟨𝑣, 0R⟩ ∈ ℝ)
9897adantl 481 . . . . . 6 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣R) → ⟨𝑣, 0R⟩ ∈ ℝ)
99 breq1 4688 . . . . . . . . . . 11 (𝑥 = ⟨𝑣, 0R⟩ → (𝑥 < 𝑦 ↔ ⟨𝑣, 0R⟩ < 𝑦))
10099notbid 307 . . . . . . . . . 10 (𝑥 = ⟨𝑣, 0R⟩ → (¬ 𝑥 < 𝑦 ↔ ¬ ⟨𝑣, 0R⟩ < 𝑦))
101100ralbidv 3015 . . . . . . . . 9 (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 ¬ ⟨𝑣, 0R⟩ < 𝑦))
102 breq2 4689 . . . . . . . . . . 11 (𝑥 = ⟨𝑣, 0R⟩ → (𝑦 < 𝑥𝑦 <𝑣, 0R⟩))
103102imbi1d 330 . . . . . . . . . 10 (𝑥 = ⟨𝑣, 0R⟩ → ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
104103ralbidv 3015 . . . . . . . . 9 (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 <𝑣, 0R⟩ → ∃𝑧𝐴 𝑦 < 𝑧)))
105101, 104anbi12d 747 . . . . . . . 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))))
106105rspcev 3340 . . . . . . 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)))
107106ex 449 . . . . . 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))))
10898, 107syl 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))))
10960, 95, 108syl2and 499 . . . 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))))
110109rexlimdva 3060 . . 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))))
11134, 45, 1103syld 60 . 2 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))))
1121113impia 1280 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 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948  cop 4216   class class class wbr 4685  cima 5146   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  1st c1st 7208  Rcnr 9725  0Rc0r 9726   <R cltr 9731  cr 9973   < cltrr 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-plp 9843  df-mp 9844  df-ltp 9845  df-enr 9915  df-nr 9916  df-plr 9917  df-mr 9918  df-ltr 9919  df-0r 9920  df-1r 9921  df-m1r 9922  df-r 9984  df-lt 9987
This theorem is referenced by: (None)
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