Theorem n0el3

Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021)

	Ref	Expression
Assertion	n0el3	$$⊢\neg \varnothing \in A\leftrightarrow \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		n0elim	$$⊢\neg \varnothing \in A\to \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$
2		n0eldmqseq	$$⊢\mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\to \neg \varnothing \in A$$
3	1 2	impbii	$$⊢\neg \varnothing \in A\leftrightarrow \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$