Theorem bnj1138

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

	Ref	Expression
Hypothesis	bnj1138.1	⊢ 𝐴 = ( 𝐵 ∪ 𝐶 )
Assertion	bnj1138	⊢ ( 𝑋 ∈ 𝐴 ↔ ( 𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1138.1	⊢ 𝐴 = ( 𝐵 ∪ 𝐶 )
2	1	eleq2i	⊢ ( 𝑋 ∈ 𝐴 ↔ 𝑋 ∈ ( 𝐵 ∪ 𝐶 ) )
3		elun	⊢ ( 𝑋 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶 ) )
4	2 3	bitri	⊢ ( 𝑋 ∈ 𝐴 ↔ ( 𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶 ) )