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Metamath Proof Explorer

Theorem snwf

Description: A singleton is well-founded if its element is. (Contributed by Mario Carneiro, 10-Jun-2013) (Revised by Mario Carneiro, 16-Nov-2014)

		Ref	Expression
	Assertion	snwf	$⊢ A \in ⋃ R_{1} [On] \to \{A\} \in ⋃ R_{1} [On]$

Step	Hyp	Ref	Expression
1		pwwf	$⊢ A \in ⋃ R_{1} [On] \leftrightarrow 𝒫 A \in ⋃ R_{1} [On]$
2		snsspw	$⊢ \{A\} \subseteq 𝒫 A$
3		sswf	$⊢ (𝒫 A \in ⋃ R_{1} [On] \land \{A\} \subseteq 𝒫 A) \to \{A\} \in ⋃ R_{1} [On]$
4	2 3	mpan2	$⊢ 𝒫 A \in ⋃ R_{1} [On] \to \{A\} \in ⋃ R_{1} [On]$
5	1 4	sylbi	$⊢ A \in ⋃ R_{1} [On] \to \{A\} \in ⋃ R_{1} [On]$