Theorem intcld 23176

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 5023	. 2 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
2		dfss3 3925	. . 3 ⊢ (𝐴 ⊆ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽))
3		iincld 23175	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽))
4	2, 3	sylan2b 605	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽))
5	1, 4	eqeltrid 2865	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝐴 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077   ⊆ wss 3904  ∅c0 4285  ∩ cint 4911  ∩ ciin 4956  ‘cfv 6536  Clsdccld 23152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-top 23030  df-cld 23155

This theorem is referenced by:  incld  23179  clscld  23183  cldmre  23214