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Theorem supsr 10523
 Description: A nonempty, bounded set of signed reals has a supremum. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
supsr ((𝐴 ≠ ∅ ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem supsr
Dummy variables 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4314 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑢 𝑢𝐴)
2 ltrelsr 10479 . . . . . . . . . . . . 13 <R ⊆ (R × R)
32brel 5616 . . . . . . . . . . . 12 (𝑦 <R 𝑥 → (𝑦R𝑥R))
43simpld 495 . . . . . . . . . . 11 (𝑦 <R 𝑥𝑦R)
54ralimi 3165 . . . . . . . . . 10 (∀𝑦𝐴 𝑦 <R 𝑥 → ∀𝑦𝐴 𝑦R)
6 dfss3 3960 . . . . . . . . . 10 (𝐴R ↔ ∀𝑦𝐴 𝑦R)
75, 6sylibr 235 . . . . . . . . 9 (∀𝑦𝐴 𝑦 <R 𝑥𝐴R)
87sseld 3970 . . . . . . . 8 (∀𝑦𝐴 𝑦 <R 𝑥 → (𝑢𝐴𝑢R))
98rexlimivw 3287 . . . . . . 7 (∃𝑥R𝑦𝐴 𝑦 <R 𝑥 → (𝑢𝐴𝑢R))
109impcom 408 . . . . . 6 ((𝑢𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → 𝑢R)
11 eleq1 2905 . . . . . . . . 9 (𝑢 = if(𝑢R, 𝑢, 1R) → (𝑢𝐴 ↔ if(𝑢R, 𝑢, 1R) ∈ 𝐴))
1211anbi1d 629 . . . . . . . 8 (𝑢 = if(𝑢R, 𝑢, 1R) → ((𝑢𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) ↔ (if(𝑢R, 𝑢, 1R) ∈ 𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)))
1312imbi1d 343 . . . . . . 7 (𝑢 = if(𝑢R, 𝑢, 1R) → (((𝑢𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧))) ↔ ((if(𝑢R, 𝑢, 1R) ∈ 𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))))
14 opeq1 4802 . . . . . . . . . . . 12 (𝑣 = 𝑤 → ⟨𝑣, 1P⟩ = ⟨𝑤, 1P⟩)
1514eceq1d 8318 . . . . . . . . . . 11 (𝑣 = 𝑤 → [⟨𝑣, 1P⟩] ~R = [⟨𝑤, 1P⟩] ~R )
1615oveq2d 7164 . . . . . . . . . 10 (𝑣 = 𝑤 → (if(𝑢R, 𝑢, 1R) +R [⟨𝑣, 1P⟩] ~R ) = (if(𝑢R, 𝑢, 1R) +R [⟨𝑤, 1P⟩] ~R ))
1716eleq1d 2902 . . . . . . . . 9 (𝑣 = 𝑤 → ((if(𝑢R, 𝑢, 1R) +R [⟨𝑣, 1P⟩] ~R ) ∈ 𝐴 ↔ (if(𝑢R, 𝑢, 1R) +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴))
1817cbvabv 2894 . . . . . . . 8 {𝑣 ∣ (if(𝑢R, 𝑢, 1R) +R [⟨𝑣, 1P⟩] ~R ) ∈ 𝐴} = {𝑤 ∣ (if(𝑢R, 𝑢, 1R) +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}
19 1sr 10492 . . . . . . . . 9 1RR
2019elimel 4537 . . . . . . . 8 if(𝑢R, 𝑢, 1R) ∈ R
2118, 20supsrlem 10522 . . . . . . 7 ((if(𝑢R, 𝑢, 1R) ∈ 𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
2213, 21dedth 4526 . . . . . 6 (𝑢R → ((𝑢𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧))))
2310, 22mpcom 38 . . . . 5 ((𝑢𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
2423ex 413 . . . 4 (𝑢𝐴 → (∃𝑥R𝑦𝐴 𝑦 <R 𝑥 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧))))
2524exlimiv 1924 . . 3 (∃𝑢 𝑢𝐴 → (∃𝑥R𝑦𝐴 𝑦 <R 𝑥 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧))))
261, 25sylbi 218 . 2 (𝐴 ≠ ∅ → (∃𝑥R𝑦𝐴 𝑦 <R 𝑥 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧))))
2726imp 407 1 ((𝐴 ≠ ∅ ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2804   ≠ wne 3021  ∀wral 3143  ∃wrex 3144   ⊆ wss 3940  ∅c0 4295  ifcif 4470  ⟨cop 4570   class class class wbr 5063  (class class class)co 7148  [cec 8277  1Pc1p 10271   ~R cer 10275  Rcnr 10276  1Rc1r 10278   +R cplr 10280
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