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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscnrm3rlem5 | Structured version Visualization version GIF version |
Description: Lemma for iscnrm3rlem6 48815. (Contributed by Zhi Wang, 5-Sep-2024.) |
Ref | Expression |
---|---|
iscnrm3rlem4.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
iscnrm3rlem4.2 | ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
iscnrm3rlem5.3 | ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) |
Ref | Expression |
---|---|
iscnrm3rlem5 | ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscnrm3rlem4.1 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) | |
2 | iscnrm3rlem4.2 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) | |
3 | eqid 2736 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | clscld 23045 | . . . 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 23045 | . . . 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 23041 | . . 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 23029 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 ∖ cdif 3947 ∩ cin 3949 ⊆ wss 3950 ∪ cuni 4905 ‘cfv 6559 Topctop 22889 Clsdccld 23014 clsccl 23016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-top 22890 df-cld 23017 df-cls 23019 |
This theorem is referenced by: iscnrm3rlem6 48815 |
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