Theorem iscld2 20832

Description: A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.)

Hypothesis

Ref	Expression
iscld.1	|- X = U. J

Assertion

Ref	Expression
iscld2	|- ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( X \ S ) e. J ) )

Proof of Theorem iscld2

Step	Hyp	Ref	Expression
1		iscld.1	. . 3 |- X = U. J
2	1	iscld 20831	. 2 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )
3	2	baibd 948	1 |- ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( X \ S ) e. J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   U.cuni 4436   ` cfv 5888   Topctop 20698   Clsdccld 20820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-cld 20823

This theorem is referenced by:  isopn2  20836  0cld  20842  uncld  20845  isclo  20891  cnclima  21072  ist1-2  21151  hausdiag  21448  qtopcld  21516  ufildr  21735  blcld  22310  icccld  22570  iocmnfcld  22572  zcld  22616  recld2  22617  qtophaus  29903  kelac2  37635  stoweidlem50  40267