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Theorem intcld 21067
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 4727 . 2 𝐴 = 𝑥𝐴 𝑥
2 dfss3 3734 . . 3 (𝐴 ⊆ (Clsd‘𝐽) ↔ ∀𝑥𝐴 𝑥 ∈ (Clsd‘𝐽))
3 iincld 21066 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑥 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝑥 ∈ (Clsd‘𝐽))
42, 3sylan2b 493 . 2 ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝑥𝐴 𝑥 ∈ (Clsd‘𝐽))
51, 4syl5eqel 2844 1 ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2140  wne 2933  wral 3051  wss 3716  c0 4059   cint 4628   ciin 4674  cfv 6050  Clsdccld 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fn 6053  df-fv 6058  df-top 20922  df-cld 21046
This theorem is referenced by:  incld  21070  clscld  21074  cldmre  21105
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