Theorem iscnrm3rlem4 48924

Description: Lemma for iscnrm3rlem8 48928. 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 4253	. . . . 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 4085	. . . . 5 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ (𝑆 ∩ 𝑇)) = ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅))
5		dif0 4337	. . . . 5 ⊢ ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅) = (𝑆 ∩ ((cls‘𝐽)‘𝑆))
6	4, 5	eqtrdi 2780	. . . 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 2729	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	sscls 22976	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
11	7, 8, 10	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
12		dfss2 3929	. . . . 5 ⊢ (𝑆 ⊆ ((cls‘𝐽)‘𝑆) ↔ (𝑆 ∩ ((cls‘𝐽)‘𝑆)) = 𝑆)
13	11, 12	sylib 218	. . . 4 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑆)) = 𝑆)
14	2, 6, 13	3eqtrd 2768	. . 3 ⊢ (𝜑 → (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = 𝑆)
15		dfss2 3929	. . 3 ⊢ (𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∖ 𝑇) ↔ (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = 𝑆)
16	14, 15	sylibr 234	. 2 ⊢ (𝜑 → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∖ 𝑇))
17		iscnrm3rlem4.4	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)
18	16, 17	sstrd 3954	1 ⊢ (𝜑 → 𝑆 ⊆ 𝑁)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ∪ cuni 4867  ‘cfv 6499  Topctop 22813  clsccl 22938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-top 22814  df-cld 22939  df-cls 22941

This theorem is referenced by:  iscnrm3rlem8  48928