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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscnrm3rlem1 | Structured version Visualization version GIF version |
Description: Lemma for iscnrm3rlem2 45937. The hypothesis could be generalized to (𝜑 → (𝑆 ∖ 𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.) |
Ref | Expression |
---|---|
iscnrm3rlem1.1 | ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
Ref | Expression |
---|---|
iscnrm3rlem1 | ⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difindi 4211 | . . . 4 ⊢ (𝑋 ∖ (𝑆 ∩ 𝑇)) = ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇)) | |
2 | 1 | ineq2i 4139 | . . 3 ⊢ (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))) = (𝑆 ∩ ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇))) |
3 | indi 4203 | . . 3 ⊢ (𝑆 ∩ ((𝑋 ∖ 𝑆) ∪ (𝑋 ∖ 𝑇))) = ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) | |
4 | disjdif 4401 | . . . . 5 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑆)) = ∅ | |
5 | 4 | uneq1i 4088 | . . . 4 ⊢ ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = (∅ ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) |
6 | 0un 4322 | . . . 4 ⊢ (∅ ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = (𝑆 ∩ (𝑋 ∖ 𝑇)) | |
7 | indif2 4200 | . . . 4 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑇)) = ((𝑆 ∩ 𝑋) ∖ 𝑇) | |
8 | 5, 6, 7 | 3eqtri 2770 | . . 3 ⊢ ((𝑆 ∩ (𝑋 ∖ 𝑆)) ∪ (𝑆 ∩ (𝑋 ∖ 𝑇))) = ((𝑆 ∩ 𝑋) ∖ 𝑇) |
9 | 2, 3, 8 | 3eqtri 2770 | . 2 ⊢ (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))) = ((𝑆 ∩ 𝑋) ∖ 𝑇) |
10 | iscnrm3rlem1.1 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
11 | df-ss 3898 | . . . 4 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆) | |
12 | 10, 11 | sylib 221 | . . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑆) |
13 | 12 | difeq1d 4051 | . 2 ⊢ (𝜑 → ((𝑆 ∩ 𝑋) ∖ 𝑇) = (𝑆 ∖ 𝑇)) |
14 | 9, 13 | eqtr2id 2792 | 1 ⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇)))) |
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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