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Theorem intcld 20654
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.
StepHypRef Expression
1 intiin 4510 . 2 𝐴 = 𝑥𝐴 𝑥
2 dfss3 3558 . . 3 (𝐴 ⊆ (Clsd‘𝐽) ↔ ∀𝑥𝐴 𝑥 ∈ (Clsd‘𝐽))
3 iincld 20653 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑥 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝑥 ∈ (Clsd‘𝐽))
42, 3sylan2b 491 . 2 ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝑥𝐴 𝑥 ∈ (Clsd‘𝐽))
51, 4syl5eqel 2692 1 ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  wne 2780  wral 2896  wss 3540  c0 3874   cint 4410   ciin 4456  cfv 5804  Clsdccld 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-top 20521  df-cld 20633
This theorem is referenced by:  incld  20657  clscld  20661  cldmre  20692
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