Theorem iscnrm3rlem4 50051

Description: Lemma for iscnrm3rlem8 50055. 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 4239	. . . . 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 4063	. . . . 5 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ (𝑆 ∩ 𝑇)) = ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅))
5		dif0 4326	. . . . 5 ⊢ ((𝑆 ∩ ((cls‘𝐽)‘𝑆)) ∖ ∅) = (𝑆 ∩ ((cls‘𝐽)‘𝑆))
6	4, 5	eqtrdi 2811	. . . 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 2760	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	sscls 23374	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
11	7, 8, 10	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
12		dfss2 4094	. . . . 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 2799	. . 3 ⊢ (𝜑 → (𝑆 ∩ (((cls‘𝐽)‘𝑆) ∖ 𝑇)) = 𝑆)
15		dfss2 4094	. . 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 3918	1 ⊢ (𝜑 → 𝑆 ⊆ 𝑁)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  ∪ cuni 4866  ‘cfv 6538  Topctop 23211  clsccl 23336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23212  df-cld 23337  df-cls 23339

This theorem is used by:  iscnrm3rlem8  50055