Theorem sepdisj 49039

Description: Separated sets are disjoint. Note that in general separatedness also requires 𝑇 ⊆ ∪ 𝐽 and (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ as well but they are unnecessary here. (Contributed by Zhi Wang, 7-Sep-2024.)

Hypotheses

Ref	Expression
sepdisj.1	⊢ (𝜑 → 𝐽 ∈ Top)
sepdisj.2	⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
sepdisj.3	⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)

Assertion

Ref	Expression
sepdisj	⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)

Proof of Theorem sepdisj

Step	Hyp	Ref	Expression
1		sepdisj.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
2		sepdisj.2	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
3		eqid 2733	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	sscls 22981	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
5	1, 2, 4	syl2anc 584	. 2 ⊢ (𝜑 → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
6		sepdisj.3	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
7	5, 6	ssdisjd 48922	1 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ∩ cin 3898   ⊆ wss 3899  ∅c0 4284  ∪ cuni 4860  ‘cfv 6489  Topctop 22818  clsccl 22943

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-top 22819  df-cld 22944  df-cls 22946

This theorem is referenced by: (None)