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Theorem suplocexpr 7526
 Description: An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m (𝜑 → ∃𝑥 𝑥𝐴)
suplocexpr.ub (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
suplocexpr.loc (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
Assertion
Ref Expression
suplocexpr (𝜑 → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))
Distinct variable groups:   𝑦,𝐴,𝑧,𝑥   𝜑,𝑦,𝑧,𝑥

Proof of Theorem suplocexpr
Dummy variables 𝑎 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocexpr.m . . 3 (𝜑 → ∃𝑥 𝑥𝐴)
2 suplocexpr.ub . . 3 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
3 suplocexpr.loc . . 3 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
4 breq1 3927 . . . . . 6 (𝑎 = 𝑤 → (𝑎 <Q 𝑢𝑤 <Q 𝑢))
54cbvrexv 2653 . . . . 5 (∃𝑎 (2nd𝐴)𝑎 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢)
65rabbii 2667 . . . 4 {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢} = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}
76opeq2i 3704 . . 3 (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
81, 2, 3, 7suplocexprlemex 7523 . 2 (𝜑 → ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ ∈ P)
91, 2, 3, 7suplocexprlemub 7524 . 2 (𝜑 → ∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦)
101, 2, 3, 7suplocexprlemlub 7525 . . 3 (𝜑 → (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))
1110ralrimivw 2504 . 2 (𝜑 → ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))
12 breq1 3927 . . . . . 6 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (𝑥<P 𝑦 ↔ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
1312notbid 656 . . . . 5 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (¬ 𝑥<P 𝑦 ↔ ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
1413ralbidv 2435 . . . 4 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ↔ ∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
15 breq2 3928 . . . . . 6 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (𝑦<P 𝑥𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩))
1615imbi1d 230 . . . . 5 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ((𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧) ↔ (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧)))
1716ralbidv 2435 . . . 4 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧) ↔ ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧)))
1814, 17anbi12d 464 . . 3 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ((∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)) ↔ (∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦 ∧ ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))))
1918rspcev 2784 . 2 ((⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ ∈ P ∧ (∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦 ∧ ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))) → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))
208, 9, 11, 19syl12anc 1214 1 (𝜑 → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 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  ⟨cop 3525  ∪ cuni 3731  ∩ cint 3766   class class class wbr 3924   “ cima 4537  1st c1st 6029  2nd c2nd 6030  Qcnq 7081
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