Theorem epini

Description: Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013)

	Ref	Expression
Hypothesis	epini.1	$$⊢A\in \mathrm{V}$$
Assertion	epini	$$⊢{\mathrm{E}}^{-1}\left[\left\{A\right\}\right]=A$$

Proof

Step	Hyp	Ref	Expression
1		epini.1	$$⊢A\in \mathrm{V}$$
2		epin	$$⊢A\in \mathrm{V}\to {\mathrm{E}}^{-1}\left[\left\{A\right\}\right]=A$$
3	1 2	ax-mp	$$⊢{\mathrm{E}}^{-1}\left[\left\{A\right\}\right]=A$$