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Metamath Proof Explorer

Theorem prtlem400

Description: Lemma for prter2 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010) (Revised by Mario Carneiro, 12-Aug-2015)

		Ref	Expression
	Hypothesis	prtlem13.1	$⊢ \underset{˙}{\sim} = \{⟨x, y⟩ \| \exists u \in A (x \in u \land y \in u)\}$
	Assertion	prtlem400	$⊢ \neg \emptyset \in (⋃ A / \underset{˙}{\sim})$

Proof

Step	Hyp	Ref	Expression
1		prtlem13.1	$⊢ \underset{˙}{\sim} = \{⟨x, y⟩ \| \exists u \in A (x \in u \land y \in u)\}$
2		neirr	$⊢ \neg \emptyset \neq \emptyset$
3	1	prtlem16	$⊢ dom \underset{˙}{\sim} = ⋃ A$
4		elqsn0	$⊢ (dom \underset{˙}{\sim} = ⋃ A \land \emptyset \in (⋃ A / \underset{˙}{\sim})) \to \emptyset \neq \emptyset$
5	3 4	mpan	$⊢ \emptyset \in (⋃ A / \underset{˙}{\sim}) \to \emptyset \neq \emptyset$
6	2 5	mto	$⊢ \neg \emptyset \in (⋃ A / \underset{˙}{\sim})$