Theorem difopn 23228

Description: The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)

Hypothesis

Ref	Expression
iscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
difopn	⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽)

Proof of Theorem difopn

Step	Hyp	Ref	Expression
1		elssuni 4909	. . . . . 6 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2		iscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	sseqtrrdi 3981	. . . . 5 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋)
4	3	adantr 486	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ⊆ 𝑋)
5		dfss2 3926	. . . 4 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∩ 𝑋) = 𝐴)
6	4, 5	sylib 221	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝑋) = 𝐴)
7	6	difeq1d 4083	. 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) = (𝐴 ∖ 𝐵))
8		indif2 4237	. . 3 ⊢ (𝐴 ∩ (𝑋 ∖ 𝐵)) = ((𝐴 ∩ 𝑋) ∖ 𝐵)
9		cldrcl 23220	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
10	9	adantl 487	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
11		simpl 488	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ 𝐽)
12	2	cldopn 23225	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽)
13	12	adantl 487	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝐵) ∈ 𝐽)
14		inopn 23093	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽)
15	10, 11, 13, 14	syl3anc 1398	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽)
16	8, 15	eqeltrrid 2871	. 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) ∈ 𝐽)
17	7, 16	eqeltrrd 2867	1 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908  ∪ cuni 4877  ‘cfv 6543  Topctop 23087  Clsdccld 23210

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fn 6546  df-fv 6551  df-top 23088  df-cld 23213

This theorem is used by:  bcthlem5  25524  cldssbrsiga  34609  pibt2  38104  poimirlem30  38342  dirkercncflem2  46859  fourierdlem62  46923