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Theorem uniopn 12359
 Description: The union of a subset of a topology (that is, the union of any family of open sets 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 12357 . . . . 5
21ibi 175 . . . 4
32simpld 111 . . 3
4 elpw2g 4117 . . . . . . . 8
54biimpar 295 . . . . . . 7
6 sseq1 3151 . . . . . . . . 9
7 unieq 3781 . . . . . . . . . 10
87eleq1d 2226 . . . . . . . . 9
96, 8imbi12d 233 . . . . . . . 8
109spcgv 2799 . . . . . . 7
115, 10syl 14 . . . . . 6
1211com23 78 . . . . 5
1312ex 114 . . . 4
1413pm2.43d 50 . . 3
153, 14mpid 42 . 2
1615imp 123 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103  wal 1333   wceq 1335   wcel 2128  wral 2435   cin 3101   wss 3102  cpw 3543  cuni 3772  ctop 12355 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545  df-uni 3773  df-top 12356 This theorem is referenced by:  iunopn  12360  unopn  12363  0opn  12364  topopn  12366  tgtop  12428  ntropn  12477  neipsm  12514  unimopn  12846  metrest  12866  cnopncntop  12897
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