Theorem iscnrm3rlem1

Description: Lemma for iscnrm3rlem2 . The hypothesis could be generalized to ( ph -> ( S \ T ) C_ X ) . (Contributed by Zhi Wang, 5-Sep-2024)

	Ref	Expression
Hypothesis	iscnrm3rlem1.1	$$⊢\phi \to S\subseteq X$$
Assertion	iscnrm3rlem1	$$⊢\phi \to S\setminus T=S\cap \left(X\setminus \left(S\cap T\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		iscnrm3rlem1.1	$$⊢\phi \to S\subseteq X$$
2		difindi	$$⊢X\setminus \left(S\cap T\right)=\left(X\setminus S\right)\cup \left(X\setminus T\right)$$
3	2	ineq2i	$$⊢S\cap \left(X\setminus \left(S\cap T\right)\right)=S\cap \left(\left(X\setminus S\right)\cup \left(X\setminus T\right)\right)$$
4		indi	$$⊢S\cap \left(\left(X\setminus S\right)\cup \left(X\setminus T\right)\right)=\left(S\cap \left(X\setminus S\right)\right)\cup \left(S\cap \left(X\setminus T\right)\right)$$
5		disjdif	$$⊢S\cap \left(X\setminus S\right)=\varnothing$$
6	5	uneq1i	$$⊢\left(S\cap \left(X\setminus S\right)\right)\cup \left(S\cap \left(X\setminus T\right)\right)=\varnothing \cup \left(S\cap \left(X\setminus T\right)\right)$$
7		0un	$$⊢\varnothing \cup \left(S\cap \left(X\setminus T\right)\right)=S\cap \left(X\setminus T\right)$$
8		indif2	$$⊢S\cap \left(X\setminus T\right)=\left(S\cap X\right)\setminus T$$
9	6 7 8	3eqtri	$$⊢\left(S\cap \left(X\setminus S\right)\right)\cup \left(S\cap \left(X\setminus T\right)\right)=\left(S\cap X\right)\setminus T$$
10	3 4 9	3eqtri	$$⊢S\cap \left(X\setminus \left(S\cap T\right)\right)=\left(S\cap X\right)\setminus T$$
11		df-ss	$$⊢S\subseteq X\leftrightarrow S\cap X=S$$
12	1 11	sylib	$$⊢\phi \to S\cap X=S$$
13	12	difeq1d	$$⊢\phi \to \left(S\cap X\right)\setminus T=S\setminus T$$
14	10 13	syl5req	$$⊢\phi \to S\setminus T=S\cap \left(X\setminus \left(S\cap T\right)\right)$$