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Theorem tgdif0 19661
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 3739 . . . . . . 7  |-  ( ( B  \  { (/) } )  i^i  ~P x
)  =  ( ( B  i^i  ~P x
)  \  { (/) } )
21unieqi 4244 . . . . . 6  |-  U. (
( B  \  { (/)
} )  i^i  ~P x )  =  U. ( ( B  i^i  ~P x )  \  { (/)
} )
3 unidif0 4610 . . . . . 6  |-  U. (
( B  i^i  ~P x )  \  { (/)
} )  =  U. ( B  i^i  ~P x
)
42, 3eqtri 2483 . . . . 5  |-  U. (
( B  \  { (/)
} )  i^i  ~P x )  =  U. ( B  i^i  ~P x
)
54sseq2i 3514 . . . 4  |-  ( x 
C_  U. ( ( B 
\  { (/) } )  i^i  ~P x )  <-> 
x  C_  U. ( B  i^i  ~P x ) )
65abbii 2588 . . 3  |-  { x  |  x  C_  U. (
( B  \  { (/)
} )  i^i  ~P x ) }  =  { x  |  x  C_ 
U. ( B  i^i  ~P x ) }
7 difexg 4585 . . . 4  |-  ( B  e.  _V  ->  ( B  \  { (/) } )  e.  _V )
8 tgval 19623 . . . 4  |-  ( ( B  \  { (/) } )  e.  _V  ->  (
topGen `  ( B  \  { (/) } ) )  =  { x  |  x  C_  U. (
( B  \  { (/)
} )  i^i  ~P x ) } )
97, 8syl 16 . . 3  |-  ( B  e.  _V  ->  ( topGen `
 ( B  \  { (/) } ) )  =  { x  |  x  C_  U. (
( B  \  { (/)
} )  i^i  ~P x ) } )
10 tgval 19623 . . 3  |-  ( B  e.  _V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
116, 9, 103eqtr4a 2521 . 2  |-  ( B  e.  _V  ->  ( topGen `
 ( B  \  { (/) } ) )  =  ( topGen `  B
) )
12 difsnexi 6581 . . . . 5  |-  ( ( B  \  { (/) } )  e.  _V  ->  B  e.  _V )
1312con3i 135 . . . 4  |-  ( -.  B  e.  _V  ->  -.  ( B  \  { (/)
} )  e.  _V )
14 fvprc 5842 . . . 4  |-  ( -.  ( B  \  { (/)
} )  e.  _V  ->  ( topGen `  ( B  \  { (/) } ) )  =  (/) )
1513, 14syl 16 . . 3  |-  ( -.  B  e.  _V  ->  (
topGen `  ( B  \  { (/) } ) )  =  (/) )
16 fvprc 5842 . . 3  |-  ( -.  B  e.  _V  ->  (
topGen `  B )  =  (/) )
1715, 16eqtr4d 2498 . 2  |-  ( -.  B  e.  _V  ->  (
topGen `  ( B  \  { (/) } ) )  =  ( topGen `  B
) )
1811, 17pm2.61i 164 1  |-  ( topGen `  ( B  \  { (/)
} ) )  =  ( topGen `  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    e. wcel 1823   {cab 2439   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   U.cuni 4235   ` cfv 5570   topGenctg 14927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-topgen 14933
This theorem is referenced by:  prdsxmslem2  21198
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