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Metamath Proof Explorer

Theorem ndmima

Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998) (Proof shortened by OpenAI, 3-Jul-2020)

		Ref	Expression
	Assertion	ndmima	$⊢ \neg A \in dom B \to B [\{A\}] = \emptyset$

Step	Hyp	Ref	Expression
1		imadisj	$⊢ B [\{A\}] = \emptyset \leftrightarrow dom B \cap \{A\} = \emptyset$
2		disjsn	$⊢ dom B \cap \{A\} = \emptyset \leftrightarrow \neg A \in dom B$
3	1 2	sylbbr	$⊢ \neg A \in dom B \to B [\{A\}] = \emptyset$