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Theorem iscnrm3rlem1 45936
Description: Lemma for iscnrm3rlem2 45937. The hypothesis could be generalized to (𝜑 → (𝑆𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.)
Hypothesis
Ref Expression
iscnrm3rlem1.1 (𝜑𝑆𝑋)
Assertion
Ref Expression
iscnrm3rlem1 (𝜑 → (𝑆𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆𝑇))))

Proof of Theorem iscnrm3rlem1
StepHypRef Expression
1 difindi 4211 . . . 4 (𝑋 ∖ (𝑆𝑇)) = ((𝑋𝑆) ∪ (𝑋𝑇))
21ineq2i 4139 . . 3 (𝑆 ∩ (𝑋 ∖ (𝑆𝑇))) = (𝑆 ∩ ((𝑋𝑆) ∪ (𝑋𝑇)))
3 indi 4203 . . 3 (𝑆 ∩ ((𝑋𝑆) ∪ (𝑋𝑇))) = ((𝑆 ∩ (𝑋𝑆)) ∪ (𝑆 ∩ (𝑋𝑇)))
4 disjdif 4401 . . . . 5 (𝑆 ∩ (𝑋𝑆)) = ∅
54uneq1i 4088 . . . 4 ((𝑆 ∩ (𝑋𝑆)) ∪ (𝑆 ∩ (𝑋𝑇))) = (∅ ∪ (𝑆 ∩ (𝑋𝑇)))
6 0un 4322 . . . 4 (∅ ∪ (𝑆 ∩ (𝑋𝑇))) = (𝑆 ∩ (𝑋𝑇))
7 indif2 4200 . . . 4 (𝑆 ∩ (𝑋𝑇)) = ((𝑆𝑋) ∖ 𝑇)
85, 6, 73eqtri 2770 . . 3 ((𝑆 ∩ (𝑋𝑆)) ∪ (𝑆 ∩ (𝑋𝑇))) = ((𝑆𝑋) ∖ 𝑇)
92, 3, 83eqtri 2770 . 2 (𝑆 ∩ (𝑋 ∖ (𝑆𝑇))) = ((𝑆𝑋) ∖ 𝑇)
10 iscnrm3rlem1.1 . . . 4 (𝜑𝑆𝑋)
11 df-ss 3898 . . . 4 (𝑆𝑋 ↔ (𝑆𝑋) = 𝑆)
1210, 11sylib 221 . . 3 (𝜑 → (𝑆𝑋) = 𝑆)
1312difeq1d 4051 . 2 (𝜑 → ((𝑆𝑋) ∖ 𝑇) = (𝑆𝑇))
149, 13eqtr2id 2792 1 (𝜑 → (𝑆𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  cdif 3878  cun 3879  cin 3880  wss 3881  c0 4252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253
This theorem is referenced by:  iscnrm3rlem2  45937
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