Theorem ntrneircomplex

Description: The relative complement of the class S exists as a subset of the base set. (Contributed by RP, 26-Jun-2021)

	Ref	Expression
Hypotheses	ntrnei.o	$$⊢O=\left(i\in \mathrm{V},j\in \mathrm{V}⟼\left(k\in \left({𝒫j}^i\right)⟼\left(l\in j⟼\left\{m\in i|l\in k\left(m\right)\right\}\right)\right)\right)$$
	ntrnei.f	$$⊢F=𝒫BOB$$
	ntrnei.r	$$⊢\phi \to IFN$$
Assertion	ntrneircomplex	$$⊢\phi \to B\setminus S\in 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		ntrnei.o	$$⊢O=\left(i\in \mathrm{V},j\in \mathrm{V}⟼\left(k\in \left({𝒫j}^i\right)⟼\left(l\in j⟼\left\{m\in i|l\in k\left(m\right)\right\}\right)\right)\right)$$
2		ntrnei.f	$$⊢F=𝒫BOB$$
3		ntrnei.r	$$⊢\phi \to IFN$$
4	1 2 3	ntrneibex	$$⊢\phi \to B\in \mathrm{V}$$
5		difssd	$$⊢\phi \to B\setminus S\subseteq B$$
6	4 5	sselpwd	$$⊢\phi \to B\setminus S\in 𝒫B$$