Theorem iscnrm3rlem4 49406

Description: Lemma for iscnrm3rlem8 49410. 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 4224	. . . . 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 4059	. . . . 5 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ (𝑆 ∩ 𝑇)) = ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅))
5		dif0 4308	. . . . 5 ⊢ ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅) = (𝑆 ∩ ((cls‘𝐽)‘𝑆))
6	4, 5	eqtrdi 2786	. . . 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 2735	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	sscls 23009	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
11	7, 8, 10	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
12		dfss2 3903	. . . . 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 2774	. . 3 ⊢ (𝜑 → (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = 𝑆)
15		dfss2 3903	. . 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 3927	1 ⊢ (𝜑 → 𝑆 ⊆ 𝑁)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3882   ∩ cin 3884   ⊆ wss 3885  ∅c0 4263  ∪ cuni 4840  ‘cfv 6487  Topctop 22846  clsccl 22971

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-top 22847  df-cld 22972  df-cls 22974

This theorem is referenced by:  iscnrm3rlem8  49410