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Metamath Proof Explorer

Theorem opidg

Description: The ordered pair <. A , A >. in Kuratowski's representation. Closed form of opid . (Contributed by Peter Mazsa, 22-Jul-2019) (Avoid depending on this detail.)

		Ref	Expression
	Assertion	opidg	$⊢ A \in V \to ⟨A, A⟩ = \{\{A\}\}$

Proof

Step	Hyp	Ref	Expression
1		dfopg	$⊢ (A \in V \land A \in V) \to ⟨A, A⟩ = \{\{A\}, \{A, A\}\}$
2	1	anidms	$⊢ A \in V \to ⟨A, A⟩ = \{\{A\}, \{A, A\}\}$
3		dfsn2	$⊢ \{A\} = \{A, A\}$
4	3	eqcomi	$⊢ \{A, A\} = \{A\}$
5	4	preq2i	$⊢ \{\{A\}, \{A, A\}\} = \{\{A\}, \{A\}\}$
6		dfsn2	$⊢ \{\{A\}\} = \{\{A\}, \{A\}\}$
7	5 6	eqtr4i	$⊢ \{\{A\}, \{A, A\}\} = \{\{A\}\}$
8	2 7	eqtrdi	$⊢ A \in V \to ⟨A, A⟩ = \{\{A\}\}$