Theorem sepdisj 48721

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 2735	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	sscls 23080	. . 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 48656	1 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  ∪ cuni 4912  ‘cfv 6563  Topctop 22915  clsccl 23042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-cls 23045

This theorem is referenced by: (None)