Theorem iscnrm3rlem5 49043

Description: Lemma for iscnrm3rlem6 49044. (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 2731	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	clscld 22962	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
5	1, 2, 4	syl2anc 584	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
6		iscnrm3rlem5.3	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)
7	3	clscld 22962	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
8	1, 6, 7	syl2anc 584	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
9		incld 22958	. . 3 ⊢ ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽)) → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
10	5, 8, 9	syl2anc 584	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
11	3	cldopn 22946	. 2 ⊢ ((((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)
12	10, 11	syl 17	1 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)

Syntax hints:   → wi 4   ∈ wcel 2111   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  ∪ cuni 4856  ‘cfv 6481  Topctop 22808  Clsdccld 22931  clsccl 22933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-top 22809  df-cld 22934  df-cls 22936

This theorem is referenced by:  iscnrm3rlem6  49044