Theorem difopn 12282

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 3764	. . . . . 6 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2		iscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	sseqtrrdi 3146	. . . . 5 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋)
4	3	adantr 274	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ⊆ 𝑋)
5		df-ss 3084	. . . 4 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∩ 𝑋) = 𝐴)
6	4, 5	sylib 121	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝑋) = 𝐴)
7	6	difeq1d 3193	. 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) = (𝐴 ∖ 𝐵))
8		indif2 3320	. . 3 ⊢ (𝐴 ∩ (𝑋 ∖ 𝐵)) = ((𝐴 ∩ 𝑋) ∖ 𝐵)
9		cldrcl 12276	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
10	9	adantl 275	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
11		simpl 108	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ 𝐽)
12	2	cldopn 12281	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽)
13	12	adantl 275	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝐵) ∈ 𝐽)
14		inopn 12175	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽)
15	10, 11, 13, 14	syl3anc 1216	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽)
16	8, 15	eqeltrrid 2227	. 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) ∈ 𝐽)
17	7, 16	eqeltrrd 2217	1 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ∖ cdif 3068   ∩ cin 3070   ⊆ wss 3071  ∪ cuni 3736  ‘cfv 5123  Topctop 12169  Clsdccld 12266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-top 12170  df-cld 12269

This theorem is referenced by: (None)