Theorem eccnvepex

Description: The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023)

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

Proof

Step	Hyp	Ref	Expression
1		snex	$$⊢\left\{A\right\}\in \mathrm{V}$$
2		cnvepresex	$$⊢\left\{A\right\}\in \mathrm{V}\to {{\mathrm{E}}^{-1}\upharpoonright }_{\left\{A\right\}}\in \mathrm{V}$$
3		ecexALTV	$$⊢{{\mathrm{E}}^{-1}\upharpoonright }_{\left\{A\right\}}\in \mathrm{V}\to \left[A\right]{\mathrm{E}}^{-1}\in \mathrm{V}$$
4	1 2 3	mp2b	$$⊢\left[A\right]{\mathrm{E}}^{-1}\in \mathrm{V}$$