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Theorem inttop2 25515
 Description: The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
inttop2
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem inttop2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniopn 16643 . . . . . . 7
21ex 423 . . . . . 6
32ral2imi 2619 . . . . 5
43adantl 452 . . . 4
5 ssiin 3952 . . . 4
6 vex 2791 . . . . . 6
76uniex 4516 . . . . 5
8 eliin 3910 . . . . 5
97, 8ax-mp 8 . . . 4
104, 5, 93imtr4g 261 . . 3
1110alrimiv 1617 . 2
12 eliin 3910 . . . . . 6
136, 12ax-mp 8 . . . . 5
14 vex 2791 . . . . . 6
15 eliin 3910 . . . . . 6
1614, 15ax-mp 8 . . . . 5
17 r19.26 2675 . . . . . . 7
18 inopn 16645 . . . . . . . . . . 11
19183expib 1154 . . . . . . . . . 10
2019ral2imi 2619 . . . . . . . . 9
2120adantl 452 . . . . . . . 8
2221com12 27 . . . . . . 7
2317, 22sylbir 204 . . . . . 6
246inex1 4155 . . . . . . 7
25 eliin 3910 . . . . . . 7
2624, 25ax-mp 8 . . . . . 6
2723, 26syl6ibr 218 . . . . 5
2813, 16, 27syl2anb 465 . . . 4
2928com12 27 . . 3
3029ralrimivv 2634 . 2
31 iinexg 4171 . . 3
32 istopg 16641 . . 3
3331, 32syl 15 . 2
3411, 30, 33mpbir2and 888 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1527   wcel 1684   wne 2446  wral 2543  cvv 2788   cin 3151   wss 3152  c0 3455  cuni 3827  ciin 3906  ctop 16631 This theorem is referenced by:  inttop3  25516 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-uni 3828  df-int 3863  df-iin 3908  df-top 16636
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