Theorem iscnrm3rlem5 49705

Description: Lemma for iscnrm3rlem6 49706. (Contributed by Zhi Wang, 5-Sep-2024.)

Hypotheses

Ref	Expression
iscnrm3rlem4.1	⊢ (𝜑 → 𝐽 ∈ Top)
iscnrm3rlem4.2	⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
iscnrm3rlem5.3	⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)

Assertion

Ref	Expression
iscnrm3rlem5	⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)

Proof of Theorem iscnrm3rlem5

Step	Hyp	Ref	Expression
1		iscnrm3rlem4.1	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
2		iscnrm3rlem4.2	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
3		eqid 2763	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	clscld 23185	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
5	1, 2, 4	syl2anc 595	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
6		iscnrm3rlem5.3	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)
7	3	clscld 23185	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
8	1, 6, 7	syl2anc 595	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
9		incld 23181	. . 3 ⊢ ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽)) → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
10	5, 8, 9	syl2anc 595	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
11	3	cldopn 23169	. 2 ⊢ ((((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)
12	10, 11	syl 18	1 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)

Syntax hints:   → wi 4   ∈ wcel 2143   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∪ cuni 4873  ‘cfv 6538  Topctop 23031  Clsdccld 23154  clsccl 23156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 23032  df-cld 23157  df-cls 23159

This theorem is referenced by:  iscnrm3rlem6  49706