Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscnrm3rlem5 Structured version   Visualization version   GIF version

Theorem iscnrm3rlem5 48814
Description: Lemma for iscnrm3rlem6 48815. (Contributed by Zhi Wang, 5-Sep-2024.)
Hypotheses
Ref Expression
iscnrm3rlem4.1 (𝜑𝐽 ∈ Top)
iscnrm3rlem4.2 (𝜑𝑆 𝐽)
iscnrm3rlem5.3 (𝜑𝑇 𝐽)
Assertion
Ref Expression
iscnrm3rlem5 (𝜑 → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)

Proof of Theorem iscnrm3rlem5
StepHypRef Expression
1 iscnrm3rlem4.1 . . . 4 (𝜑𝐽 ∈ Top)
2 iscnrm3rlem4.2 . . . 4 (𝜑𝑆 𝐽)
3 eqid 2736 . . . . 5 𝐽 = 𝐽
43clscld 23045 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
51, 2, 4syl2anc 584 . . 3 (𝜑 → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
6 iscnrm3rlem5.3 . . . 4 (𝜑𝑇 𝐽)
73clscld 23045 . . . 4 ((𝐽 ∈ Top ∧ 𝑇 𝐽) → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
81, 6, 7syl2anc 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))
105, 8, 9syl2anc 584 . 2 (𝜑 → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
113cldopn 23029 . 2 ((((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽) → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)
1210, 11syl 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
  Copyright terms: Public domain W3C validator