Theorem iscnrm3rlem4 45676

Description: Lemma for iscnrm3rlem8 45680. 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 4190	. . . . 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 4030	. . . . 5 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ (𝑆 ∩ 𝑇)) = ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅))
5		dif0 4273	. . . . 5 ⊢ ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅) = (𝑆 ∩ ((cls‘𝐽)‘𝑆))
6	4, 5	eqtrdi 2809	. . . 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 2758	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	sscls 21769	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
11	7, 8, 10	syl2anc 587	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
12		df-ss 3877	. . . . 5 ⊢ (𝑆 ⊆ ((cls‘𝐽)‘𝑆) ↔ (𝑆 ∩ ((cls‘𝐽)‘𝑆)) = 𝑆)
13	11, 12	sylib 221	. . . 4 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑆)) = 𝑆)
14	2, 6, 13	3eqtrd 2797	. . 3 ⊢ (𝜑 → (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = 𝑆)
15		df-ss 3877	. . 3 ⊢ (𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∖ 𝑇) ↔ (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = 𝑆)
16	14, 15	sylibr 237	. 2 ⊢ (𝜑 → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∖ 𝑇))
17		iscnrm3rlem4.4	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)
18	16, 17	sstrd 3904	1 ⊢ (𝜑 → 𝑆 ⊆ 𝑁)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ∪ cuni 4801  ‘cfv 6340  Topctop 21606  clsccl 21731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-top 21607  df-cld 21732  df-cls 21734

This theorem is referenced by:  iscnrm3rlem8  45680