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Metamath Proof Explorer

Theorem lubeu

Description: Unique existence proper of a member of the domain of the least upper bound function of a poset. (Contributed by NM, 7-Sep-2018)

Ref Expression
Hypotheses lubval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
lubval.l ⊢ ≤ = ( le ‘ 𝐾 )
lubval.u ⊢ 𝑈 = ( lub ‘ 𝐾 )
lubval.p ⊢ ( 𝜓 ↔ ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) )
lubval.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
lubeleu.s ⊢ ( 𝜑 → 𝑆 ∈ dom 𝑈 )
Assertion lubeu ( 𝜑 → ∃! 𝑥 ∈ 𝐵 𝜓 )

Proof

Step Hyp Ref Expression
1 lubval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 lubval.l ⊢ ≤ = ( le ‘ 𝐾 )
3 lubval.u ⊢ 𝑈 = ( lub ‘ 𝐾 )
4 lubval.p ⊢ ( 𝜓 ↔ ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) )
5 lubval.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
6 lubeleu.s ⊢ ( 𝜑 → 𝑆 ∈ dom 𝑈 )
7 1 2 3 4 5 lubeldm ⊢ ( 𝜑 → ( 𝑆 ∈ dom 𝑈 ↔ ( 𝑆 ⊆ 𝐵 ∧ ∃! 𝑥 ∈ 𝐵 𝜓 ) ) )
8 6 7 mpbid ⊢ ( 𝜑 → ( 𝑆 ⊆ 𝐵 ∧ ∃! 𝑥 ∈ 𝐵 𝜓 ) )
9 8 simprd ⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐵 𝜓 )