Theorem iscnrm3rlem1 48785

Description: Lemma for iscnrm3rlem2 48786. 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 4274	. . . 4 ⊢ (𝑋 ∖ (𝑆 ∩ 𝑇)) = ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇))
2	1	ineq2i 4199	. . 3 ⊢ (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))) = (𝑆 ∩ ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇)))
3		indi 4266	. . 3 ⊢ (𝑆 ∩ ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇))) = ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇)))
4		disjdif 4454	. . . . 5 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑆)) = ∅
5	4	uneq1i 4146	. . . 4 ⊢ ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = (∅ ∪ (𝑆 ∩ (𝑋 ∖ 𝑇)))
6		0un 4378	. . . 4 ⊢ (∅ ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = (𝑆 ∩ (𝑋 ∖ 𝑇))
7		indif2 4263	. . . 4 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑇)) = ((𝑆 ∩ 𝑋) ∖ 𝑇)
8	5, 6, 7	3eqtri 2761	. . 3 ⊢ ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = ((𝑆 ∩ 𝑋) ∖ 𝑇)
9	2, 3, 8	3eqtri 2761	. 2 ⊢ (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))) = ((𝑆 ∩ 𝑋) ∖ 𝑇)
10		iscnrm3rlem1.1	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
11		dfss2 3951	. . . 4 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆)
12	10, 11	sylib 218	. . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑆)
13	12	difeq1d 4107	. 2 ⊢ (𝜑 → ((𝑆 ∩ 𝑋) ∖ 𝑇) = (𝑆 ∖ 𝑇))
14	9, 13	eqtr2id 2782	1 ⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))))

Syntax hints:   → wi 4   = wceq 1539   ∖ cdif 3930   ∪ cun 3931   ∩ cin 3932   ⊆ wss 3933  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316

This theorem is referenced by:  iscnrm3rlem2  48786