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 ) )