Theorem sepdisj 46218

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 2738	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	sscls 22207	. . 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 46153	1 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839  ‘cfv 6433  Topctop 22042  clsccl 22169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170  df-cls 22172

This theorem is referenced by: (None)