Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  is1stc Structured version   Unicode version

Theorem is1stc 17494
 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 4016 . . . 4
2 is1stc.1 . . . 4
31, 2syl6eqr 2485 . . 3
4 pweq 3794 . . . 4
5 raleq 2896 . . . . 5
65anbi2d 685 . . . 4
74, 6rexeqbidv 2909 . . 3
83, 7raleqbidv 2908 . 2
9 df-1stc 17492 . 2
108, 9elrab2 3086 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698   cin 3311  cpw 3791  cuni 4007   class class class wbr 4204  com 4837   cdom 7099  ctop 16948  c1stc 17490 This theorem is referenced by:  is1stc2  17495  1stctop  17496 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793  df-uni 4008  df-1stc 17492
 Copyright terms: Public domain W3C validator