Theorem elun

Description: Expansion of membership in class union. Theorem 12 of Suppes p. 25. (Contributed by NM, 7-Aug-1994)

	Ref	Expression
Assertion	elun	⊢ ( 𝐴 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ ( 𝐵 ∪ 𝐶 ) → 𝐴 ∈ V )
2		elex	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )
3		elex	⊢ ( 𝐴 ∈ 𝐶 → 𝐴 ∈ V )
4	2 3	jaoi	⊢ ( ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) → 𝐴 ∈ V )
5		eleq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
6		eleq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶 ) )
7	5 6	orbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶 ) ) )
8		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
9		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
10	8 9	orbi12d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) ) )
11		df-un	⊢ ( 𝐵 ∪ 𝐶 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶 ) }
12	7 10 11	elab2gw	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) ) )
13	1 4 12	pm5.21nii	⊢ ( 𝐴 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) )