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Theorem iscnrm3rlem5 48662
Description: Lemma for iscnrm3rlem6 48663. (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 2733 . . . . 5 𝐽 = 𝐽
43clscld 23052 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
51, 2, 4syl2anc 583 . . 3 (𝜑 → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
6 iscnrm3rlem5.3 . . . 4 (𝜑𝑇 𝐽)
73clscld 23052 . . . 4 ((𝐽 ∈ Top ∧ 𝑇 𝐽) → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
81, 6, 7syl2anc 583 . . 3 (𝜑 → ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽))
9 incld 23048 . . 3 ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘𝑇) ∈ (Clsd‘𝐽)) → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
105, 8, 9syl2anc 583 . 2 (𝜑 → (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘𝐽))
113cldopn 23036 . 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 2104  cdif 3960  cin 3962  wss 3963   cuni 4914  cfv 6558  Topctop 22896  Clsdccld 23021  clsccl 23023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3377  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-int 4954  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  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 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-top 22897  df-cld 23024  df-cls 23026
This theorem is referenced by:  iscnrm3rlem6  48663
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