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Metamath Proof Explorer

Theorem bnj1424

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypothesis bnj1424.1 ⊢ 𝐴 = ( 𝐵 ∪ 𝐶 )
Assertion bnj1424 ( 𝐷 ∈ 𝐴 → ( 𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 bnj1424.1 ⊢ 𝐴 = ( 𝐵 ∪ 𝐶 )
2 1 bnj1138 ⊢ ( 𝐷 ∈ 𝐴 ↔ ( 𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶 ) )
3 2 biimpi ⊢ ( 𝐷 ∈ 𝐴 → ( 𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶 ) )