Metamath Proof Explorer

Theorem joineu

Description: Uniqueness of join of elements in the domain. (Contributed by NM, 12-Sep-2018)

Ref Expression
Hypotheses joinval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
joinval2.l ⊢ ≤ = ( le ‘ 𝐾 )
joinval2.j ⊢ ∨ = ( join ‘ 𝐾 )
joinval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
joinval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
joinval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
joinlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
Assertion joineu ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 joinval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 joinval2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 joinval2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 joinval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5 joinval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 joinval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 joinlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
8 eqid ⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 )
9 8 3 4 5 6 joindef ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ↔ { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) ) )
10 biid ⊢ ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) )
11 4 adantr ⊢ ( ( 𝜑 ∧ { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) ) → 𝐾 ∈ 𝑉 )
12 simpr ⊢ ( ( 𝜑 ∧ { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) ) → { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) )
13 1 2 8 10 11 12 lubeu ⊢ ( ( 𝜑 ∧ { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) ) → ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) )
14 13 ex ⊢ ( 𝜑 → ( { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) → ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) )
15 1 2 3 4 5 6 joinval2lem ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
16 5 6 15 syl2anc ⊢ ( 𝜑 → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
17 16 reubidv ⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ∃! 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
18 14 17 sylibd ⊢ ( 𝜑 → ( { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) → ∃! 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
19 9 18 sylbid ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ → ∃! 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
20 7 19 mpd ⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) )