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Metamath Proof Explorer

Theorem iunxsn

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004) (Proof shortened by Mario Carneiro, 25-Jun-2016)

	Ref	Expression
Hypotheses	iunxsn.1	⊢ 𝐴 ∈ V
	iunxsn.2	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
Assertion	iunxsn	⊢ ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶

Proof

Step	Hyp	Ref	Expression
1		iunxsn.1	⊢ 𝐴 ∈ V
2		iunxsn.2	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
3	2	iunxsng	⊢ ( 𝐴 ∈ V → ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 )
4	1 3	ax-mp	⊢ ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶