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Theorem uniopn 16970
 Description: The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
Assertion
Ref Expression
uniopn

Proof of Theorem uniopn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istopg 16968 . . . . 5
21ibi 233 . . . 4
32simpld 446 . . 3
4 elpw2g 4363 . . . . . . . 8
54biimpar 472 . . . . . . 7
6 sseq1 3369 . . . . . . . . 9
7 unieq 4024 . . . . . . . . . 10
87eleq1d 2502 . . . . . . . . 9
96, 8imbi12d 312 . . . . . . . 8
109spcgv 3036 . . . . . . 7
115, 10syl 16 . . . . . 6
1211com23 74 . . . . 5
1312ex 424 . . . 4
1413pm2.43d 46 . . 3
153, 14mpid 39 . 2
1615imp 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549   wceq 1652   wcel 1725  wral 2705   cin 3319   wss 3320  cpw 3799  cuni 4015  ctop 16958 This theorem is referenced by:  iunopn  16971  unopn  16976  0opn  16977  topopn  16979  tgtop  17038  ntropn  17113  toponmre  17157  neips  17177  txcmplem1  17673  unimopn  18526  metrest  18554  cvmscld  24960  mblfinlem3  26245  mblfinlem4  26246  ismblfin  26247 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801  df-uni 4016  df-top 16963
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