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Theorem is1stc 20236
 Description: The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.)
Hypothesis
Ref Expression
is1stc.1
Assertion
Ref Expression
is1stc
Distinct variable groups:   ,,,   ,
Allowed substitution hints:   (,)

Proof of Theorem is1stc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 unieq 4201 . . . 4
2 is1stc.1 . . . 4
31, 2syl6eqr 2463 . . 3
4 pweq 3960 . . . 4
5 raleq 3006 . . . . 5
65anbi2d 704 . . . 4
74, 6rexeqbidv 3021 . . 3
83, 7raleqbidv 3020 . 2
9 df-1stc 20234 . 2
108, 9elrab2 3211 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 186   wa 369   wceq 1407   wcel 1844  wral 2756  wrex 2757   cin 3415  cpw 3957  cuni 4193   class class class wbr 4397  com 6685   cdom 7554  ctop 19688  c1stc 20232 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382 This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-in 3423  df-ss 3430  df-pw 3959  df-uni 4194  df-1stc 20234 This theorem is referenced by:  is1stc2  20237  1stctop  20238
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