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Theorem unint2t 25621
 Description: The intersection of two topologies over the same underlying set is a topology over . compare uniin 3863. (Contributed by FL, 27-Nov-2011.)
Assertion
Ref Expression
unint2t

Proof of Theorem unint2t
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniin 3863 . . 3
2 inss1 3402 . . . 4
32a1i 10 . . 3
41, 3syl5ss 3203 . 2
5 eqid 2296 . . . . . . . 8
65topopn 16668 . . . . . . 7
7 eqid 2296 . . . . . . . . 9
87topopn 16668 . . . . . . . 8
9 eleq1 2356 . . . . . . . . . . 11
109eqcoms 2299 . . . . . . . . . 10
11 elin 3371 . . . . . . . . . . 11
1211simplbi2com 1364 . . . . . . . . . 10
1310, 12syl6bi 219 . . . . . . . . 9
1413com3l 75 . . . . . . . 8
158, 14syl 15 . . . . . . 7
166, 15syl5com 26 . . . . . 6
17163imp 1145 . . . . 5
18 elssuni 3871 . . . . 5
19 sseq2 3213 . . . . . 6
2019rspcev 2897 . . . . 5
2117, 18, 20syl2an 463 . . . 4
2221ralrimiva 2639 . . 3
23 uniss2 3874 . . 3
2422, 23syl 15 . 2
254, 24eqssd 3209 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   w3a 934   wceq 1632   wcel 1696  wral 2556  wrex 2557   cin 3164   wss 3165  cuni 3843  ctop 16647 This theorem is referenced by:  intcont  25646 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-top 16652
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