Theorem iscnrm3rlem4 47665

Description: Lemma for iscnrm3rlem8 47669. Given two disjoint subsets 𝑆 and 𝑇 of the underlying set of a topology 𝐽, if 𝑁 is a superset of (((cls‘𝐽)‘𝑆) ∖ 𝑇), then it is a superset of 𝑆. (Contributed by Zhi Wang, 5-Sep-2024.)

Hypotheses

Ref	Expression
iscnrm3rlem4.1	⊢ (𝜑 → 𝐽 ∈ Top)
iscnrm3rlem4.2	⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
iscnrm3rlem4.3	⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
iscnrm3rlem4.4	⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)

Assertion

Ref	Expression
iscnrm3rlem4	⊢ (𝜑 → 𝑆 ⊆ 𝑁)

Proof of Theorem iscnrm3rlem4

Step	Hyp	Ref	Expression
1		indifdi 4284	. . . . 5 ⊢ (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ (𝑆 ∩ 𝑇))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ (𝑆 ∩ 𝑇)))
3		iscnrm3rlem4.3	. . . . . 6 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
4	3	difeq2d 4123	. . . . 5 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ (𝑆 ∩ 𝑇)) = ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅))
5		dif0 4373	. . . . 5 ⊢ ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅) = (𝑆 ∩ ((cls‘𝐽)‘𝑆))
6	4, 5	eqtrdi 2787	. . . 4 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ (𝑆 ∩ 𝑇)) = (𝑆 ∩ ((cls‘𝐽)‘𝑆)))
7		iscnrm3rlem4.1	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
8		iscnrm3rlem4.2	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
9		eqid 2731	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	sscls 22781	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
11	7, 8, 10	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
12		df-ss 3966	. . . . 5 ⊢ (𝑆 ⊆ ((cls‘𝐽)‘𝑆) ↔ (𝑆 ∩ ((cls‘𝐽)‘𝑆)) = 𝑆)
13	11, 12	sylib 217	. . . 4 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑆)) = 𝑆)
14	2, 6, 13	3eqtrd 2775	. . 3 ⊢ (𝜑 → (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = 𝑆)
15		df-ss 3966	. . 3 ⊢ (𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∖ 𝑇) ↔ (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = 𝑆)
16	14, 15	sylibr 233	. 2 ⊢ (𝜑 → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∖ 𝑇))
17		iscnrm3rlem4.4	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)
18	16, 17	sstrd 3993	1 ⊢ (𝜑 → 𝑆 ⊆ 𝑁)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ∖ cdif 3946   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  ∪ cuni 4909  ‘cfv 6544  Topctop 22616  clsccl 22743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22617  df-cld 22744  df-cls 22746

This theorem is referenced by:  iscnrm3rlem8  47669