Theorem iscnrm3rlem5 49256

Description: Lemma for iscnrm3rlem6 49257. (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 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	clscld 22995	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
5	1, 2, 4	syl2anc 585	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
6		iscnrm3rlem5.3	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)
7	3	clscld 22995	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
8	1, 6, 7	syl2anc 585	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
9		incld 22991	. . 3 ⊢ ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽)) → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
10	5, 8, 9	syl2anc 585	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
11	3	cldopn 22979	. 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 2114   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4864  ‘cfv 6493  Topctop 22841  Clsdccld 22964  clsccl 22966

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 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-top 22842  df-cld 22967  df-cls 22969

This theorem is referenced by:  iscnrm3rlem6  49257