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Metamath Proof Explorer

Theorem chjvali

Description: Value of join in CH . (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

Ref Expression
Hypotheses chjval.1 ⊢ 𝐴 ∈ Cℋ
chjval.2 ⊢ 𝐵 ∈ Cℋ
Assertion chjvali ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 chjval.1 ⊢ 𝐴 ∈ Cℋ
2 chjval.2 ⊢ 𝐵 ∈ Cℋ
3 chjval ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
4 1 2 3 mp2an ⊢ ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) )