Theorem reuun1

Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005)

	Ref	Expression
Assertion	reuun1	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( 𝜑 ∨ 𝜓 ) ) → ∃! 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ssun1	⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 )
2		orc	⊢ ( 𝜑 → ( 𝜑 ∨ 𝜓 ) )
3	2	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → ( 𝜑 ∨ 𝜓 ) )
4		reuss2	⊢ ( ( ( 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → ( 𝜑 ∨ 𝜓 ) ) ) ∧ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( 𝜑 ∨ 𝜓 ) ) ) → ∃! 𝑥 ∈ 𝐴 𝜑 )
5	1 3 4	mpanl12	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( 𝜑 ∨ 𝜓 ) ) → ∃! 𝑥 ∈ 𝐴 𝜑 )