Theorem 1stctop 21246

Description: A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)

Assertion

Ref	Expression
1stctop	|- ( J e. 1stc -> J e. Top )

Proof of Theorem 1stctop

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- U. J = U. J
2	1	is1stc 21244	. 2 |- ( J e. 1stc <-> ( J e. Top /\ A. x e. U. J E. y e. ~P J ( y ~<_ om /\ A. z e. J ( x e. z -> x e. U. ( y i^i ~P z ) ) ) ) )
3	2	simplbi 476	1 |- ( J e. 1stc -> J e. Top )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   omcom 7065   ~<_ cdom 7953   Topctop 20698   1stcc1stc 21240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-1stc 21242

This theorem is referenced by:  1stcfb  21248  1stcrest  21256  1stcelcls  21264  lly1stc  21299  1stckgen  21357  tx1stc  21453