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Theorem 2ndci 17190
 Description: A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
2ndci

Proof of Theorem 2ndci
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3
2 simpr 447 . . 3
3 eqidd 2297 . . 3
4 breq1 4042 . . . . 5
5 fveq2 5541 . . . . . 6
65eqeq1d 2304 . . . . 5
74, 6anbi12d 691 . . . 4
87rspcev 2897 . . 3
91, 2, 3, 8syl12anc 1180 . 2
10 is2ndc 17188 . 2
119, 10sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1632   wcel 1696  wrex 2557   class class class wbr 4039  com 4672  cfv 5271   cdom 6877  ctg 13358  ctb 16651  c2ndc 17180 This theorem is referenced by:  2ndcrest  17196  2ndcomap  17200  dis2ndc  17202  dis1stc  17241  tx2ndc  17361  met2ndci  18084  re2ndc  18323 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-nul 4165 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-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-2ndc 17182
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