Theorem intcld 23064

Description: The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)

Assertion

Ref	Expression
intcld	⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝐴 ∈ (Clsd‘𝐽))

Proof of Theorem intcld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intiin 5064	. 2 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
2		dfss3 3984	. . 3 ⊢ (𝐴 ⊆ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽))
3		iincld 23063	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽))
4	2, 3	sylan2b 594	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽))
5	1, 4	eqeltrid 2843	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝐴 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059   ⊆ wss 3963  ∅c0 4339  ∩ cint 4951  ∩ ciin 4997  ‘cfv 6563  Clsdccld 23040

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-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

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-rab 3434  df-v 3480  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-iin 4999  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-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-top 22916  df-cld 23043

This theorem is referenced by:  incld  23067  clscld  23071  cldmre  23102