Theorem iscnrm3rlem1 49853

Description: Lemma for iscnrm3rlem2 49854. The hypothesis could be generalized to (𝜑 → (𝑆 ∖ 𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.)

Hypothesis

Ref	Expression
iscnrm3rlem1.1	⊢ (𝜑 → 𝑆 ⊆ 𝑋)

Assertion

Ref	Expression
iscnrm3rlem1	⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))))

Proof of Theorem iscnrm3rlem1

Step	Hyp	Ref	Expression
1		difindi 4241	. . . 4 ⊢ (𝑋 ∖ (𝑆 ∩ 𝑇)) = ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇))
2	1	ineq2i 4166	. . 3 ⊢ (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))) = (𝑆 ∩ ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇)))
3		indi 4233	. . 3 ⊢ (𝑆 ∩ ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇))) = ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇)))
4		disjdif 4429	. . . . 5 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑆)) = ∅
5	4	uneq1i 4114	. . . 4 ⊢ ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = (∅ ∪ (𝑆 ∩ (𝑋 ∖ 𝑇)))
6		0un 4349	. . . 4 ⊢ (∅ ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = (𝑆 ∩ (𝑋 ∖ 𝑇))
7		indif2 4230	. . . 4 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑇)) = ((𝑆 ∩ 𝑋) ∖ 𝑇)
8	5, 6, 7	3eqtri 2789	. . 3 ⊢ ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = ((𝑆 ∩ 𝑋) ∖ 𝑇)
9	2, 3, 8	3eqtri 2789	. 2 ⊢ (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))) = ((𝑆 ∩ 𝑋) ∖ 𝑇)
10		iscnrm3rlem1.1	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
11		dfss2 3920	. . . 4 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆)
12	10, 11	sylib 221	. . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑆)
13	12	difeq1d 4076	. 2 ⊢ (𝜑 → ((𝑆 ∩ 𝑋) ∖ 𝑇) = (𝑆 ∖ 𝑇))
14	9, 13	eqtr2id 2810	1 ⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283

This theorem is used by:  iscnrm3rlem2  49854