Theorem sepdisj 48917

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 2730	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	sscls 22950	. . 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 48800	1 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  ∪ cuni 4874  ‘cfv 6514  Topctop 22787  clsccl 22912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-top 22788  df-cld 22913  df-cls 22915

This theorem is referenced by: (None)