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Metamath Proof Explorer

Theorem brcodir

Description: Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015)

Ref Expression
Assertion brcodir ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( ◡ 𝑅 ∘ 𝑅 ) 𝐵 ↔ ∃ 𝑧 ( 𝐴 𝑅 𝑧 ∧ 𝐵 𝑅 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 brcog ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( ◡ 𝑅 ∘ 𝑅 ) 𝐵 ↔ ∃ 𝑧 ( 𝐴 𝑅 𝑧 ∧ 𝑧 ◡ 𝑅 𝐵 ) ) )
2 vex ⊢ 𝑧 ∈ V
3 brcnvg ⊢ ( ( 𝑧 ∈ V ∧ 𝐵 ∈ 𝑊 ) → ( 𝑧 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝑧 ) )
4 2 3 mpan ⊢ ( 𝐵 ∈ 𝑊 → ( 𝑧 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝑧 ) )
5 4 anbi2d ⊢ ( 𝐵 ∈ 𝑊 → ( ( 𝐴 𝑅 𝑧 ∧ 𝑧 ◡ 𝑅 𝐵 ) ↔ ( 𝐴 𝑅 𝑧 ∧ 𝐵 𝑅 𝑧 ) ) )
6 5 adantl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 𝑅 𝑧 ∧ 𝑧 ◡ 𝑅 𝐵 ) ↔ ( 𝐴 𝑅 𝑧 ∧ 𝐵 𝑅 𝑧 ) ) )
7 6 exbidv ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑧 ( 𝐴 𝑅 𝑧 ∧ 𝑧 ◡ 𝑅 𝐵 ) ↔ ∃ 𝑧 ( 𝐴 𝑅 𝑧 ∧ 𝐵 𝑅 𝑧 ) ) )
8 1 7 bitrd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( ◡ 𝑅 ∘ 𝑅 ) 𝐵 ↔ ∃ 𝑧 ( 𝐴 𝑅 𝑧 ∧ 𝐵 𝑅 𝑧 ) ) )