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Theorem tgdif0 20018
Description: A generated topology is not affected by the addition or removal of the empty set from the base. (Contributed by Mario Carneiro, 28-Aug-2015.)
Assertion
Ref Expression
tgdif0  |-  ( topGen `  ( B  \  { (/)
} ) )  =  ( topGen `  B )

Proof of Theorem tgdif0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 indif1 3654 . . . . . . 7  |-  ( ( B  \  { (/) } )  i^i  ~P x
)  =  ( ( B  i^i  ~P x
)  \  { (/) } )
21unieqi 4176 . . . . . 6  |-  U. (
( B  \  { (/)
} )  i^i  ~P x )  =  U. ( ( B  i^i  ~P x )  \  { (/)
} )
3 unidif0 4548 . . . . . 6  |-  U. (
( B  i^i  ~P x )  \  { (/)
} )  =  U. ( B  i^i  ~P x
)
42, 3eqtri 2473 . . . . 5  |-  U. (
( B  \  { (/)
} )  i^i  ~P x )  =  U. ( B  i^i  ~P x
)
54sseq2i 3424 . . . 4  |-  ( x 
C_  U. ( ( B 
\  { (/) } )  i^i  ~P x )  <-> 
x  C_  U. ( B  i^i  ~P x ) )
65abbii 2567 . . 3  |-  { x  |  x  C_  U. (
( B  \  { (/)
} )  i^i  ~P x ) }  =  { x  |  x  C_ 
U. ( B  i^i  ~P x ) }
7 difexg 4523 . . . 4  |-  ( B  e.  _V  ->  ( B  \  { (/) } )  e.  _V )
8 tgval 19980 . . . 4  |-  ( ( B  \  { (/) } )  e.  _V  ->  (
topGen `  ( B  \  { (/) } ) )  =  { x  |  x  C_  U. (
( B  \  { (/)
} )  i^i  ~P x ) } )
97, 8syl 17 . . 3  |-  ( B  e.  _V  ->  ( topGen `
 ( B  \  { (/) } ) )  =  { x  |  x  C_  U. (
( B  \  { (/)
} )  i^i  ~P x ) } )
10 tgval 19980 . . 3  |-  ( B  e.  _V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
116, 9, 103eqtr4a 2511 . 2  |-  ( B  e.  _V  ->  ( topGen `
 ( B  \  { (/) } ) )  =  ( topGen `  B
) )
12 difsnexi 6586 . . . . 5  |-  ( ( B  \  { (/) } )  e.  _V  ->  B  e.  _V )
1312con3i 142 . . . 4  |-  ( -.  B  e.  _V  ->  -.  ( B  \  { (/)
} )  e.  _V )
14 fvprc 5841 . . . 4  |-  ( -.  ( B  \  { (/)
} )  e.  _V  ->  ( topGen `  ( B  \  { (/) } ) )  =  (/) )
1513, 14syl 17 . . 3  |-  ( -.  B  e.  _V  ->  (
topGen `  ( B  \  { (/) } ) )  =  (/) )
16 fvprc 5841 . . 3  |-  ( -.  B  e.  _V  ->  (
topGen `  B )  =  (/) )
1715, 16eqtr4d 2488 . 2  |-  ( -.  B  e.  _V  ->  (
topGen `  ( B  \  { (/) } ) )  =  ( topGen `  B
) )
1811, 17pm2.61i 169 1  |-  ( topGen `  ( B  \  { (/)
} ) )  =  ( topGen `  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1447    e. wcel 1890   {cab 2437   _Vcvv 3012    \ cdif 3368    i^i cin 3370    C_ wss 3371   (/)c0 3698   ~Pcpw 3918   {csn 3935   U.cuni 4167   ` cfv 5560   topGenctg 15346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5524  df-fun 5562  df-fv 5568  df-topgen 15352
This theorem is referenced by:  prdsxmslem2  21554
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