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		n0el2	$$⊢\neg \varnothing \in A\leftrightarrow \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$
2	1	biimpi	$$⊢\neg \varnothing \in A\to \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$
3	2	qseq1d	$$⊢\neg \varnothing \in A\to \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$
4		qsresid	$$⊢A/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A/{\mathrm{E}}^{-1}$$
5		qsid	$$⊢A/{\mathrm{E}}^{-1}=A$$
6	4 5	eqtri	$$⊢A/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$
7	3 6	syl6eq	$$⊢\neg \varnothing \in A\to \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$
8		n0eldmqseq	$$⊢\mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\to \neg \varnothing \in A$$
9	7 8	impbii	$$⊢\neg \varnothing \in A\leftrightarrow \mathrm{dom}\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A$$