Theorem sepdisj 48604

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 2740	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	sscls 23085	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
5	1, 2, 4	syl2anc 583	. 2 ⊢ (𝜑 → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
6		sepdisj.3	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
7	5, 6	ssdisjd 48539	1 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  clsccl 23047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-top 22921  df-cld 23048  df-cls 23050

This theorem is referenced by: (None)