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Theorem tgdif0 17049
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 3577 . . . . . . 7  |-  ( ( B  \  { (/) } )  i^i  ~P x
)  =  ( ( B  i^i  ~P x
)  \  { (/) } )
21unieqi 4017 . . . . . 6  |-  U. (
( B  \  { (/)
} )  i^i  ~P x )  =  U. ( ( B  i^i  ~P x )  \  { (/)
} )
3 unidif0 4364 . . . . . 6  |-  U. (
( B  i^i  ~P x )  \  { (/)
} )  =  U. ( B  i^i  ~P x
)
42, 3eqtri 2455 . . . . 5  |-  U. (
( B  \  { (/)
} )  i^i  ~P x )  =  U. ( B  i^i  ~P x
)
54sseq2i 3365 . . . 4  |-  ( x 
C_  U. ( ( B 
\  { (/) } )  i^i  ~P x )  <-> 
x  C_  U. ( B  i^i  ~P x ) )
65abbii 2547 . . 3  |-  { x  |  x  C_  U. (
( B  \  { (/)
} )  i^i  ~P x ) }  =  { x  |  x  C_ 
U. ( B  i^i  ~P x ) }
7 difexg 4343 . . . 4  |-  ( B  e.  _V  ->  ( B  \  { (/) } )  e.  _V )
8 tgval 17012 . . . 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 17012 . . 3  |-  ( B  e.  _V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
116, 9, 103eqtr4a 2493 . 2  |-  ( B  e.  _V  ->  ( topGen `
 ( B  \  { (/) } ) )  =  ( topGen `  B
) )
12 ssun1 3502 . . . . . . 7  |-  B  C_  ( B  u.  { (/) } )
13 undif1 3695 . . . . . . 7  |-  ( ( B  \  { (/) } )  u.  { (/) } )  =  ( B  u.  { (/) } )
1412, 13sseqtr4i 3373 . . . . . 6  |-  B  C_  ( ( B  \  { (/) } )  u. 
{ (/) } )
15 p0ex 4378 . . . . . . 7  |-  { (/) }  e.  _V
16 unexg 4702 . . . . . . 7  |-  ( ( ( B  \  { (/)
} )  e.  _V  /\ 
{ (/) }  e.  _V )  ->  ( ( B 
\  { (/) } )  u.  { (/) } )  e.  _V )
1715, 16mpan2 653 . . . . . 6  |-  ( ( B  \  { (/) } )  e.  _V  ->  ( ( B  \  { (/)
} )  u.  { (/)
} )  e.  _V )
18 ssexg 4341 . . . . . 6  |-  ( ( B  C_  ( ( B  \  { (/) } )  u.  { (/) } )  /\  ( ( B 
\  { (/) } )  u.  { (/) } )  e.  _V )  ->  B  e.  _V )
1914, 17, 18sylancr 645 . . . . 5  |-  ( ( B  \  { (/) } )  e.  _V  ->  B  e.  _V )
2019con3i 129 . . . 4  |-  ( -.  B  e.  _V  ->  -.  ( B  \  { (/)
} )  e.  _V )
21 fvprc 5714 . . . 4  |-  ( -.  ( B  \  { (/)
} )  e.  _V  ->  ( topGen `  ( B  \  { (/) } ) )  =  (/) )
2220, 21syl 16 . . 3  |-  ( -.  B  e.  _V  ->  (
topGen `  ( B  \  { (/) } ) )  =  (/) )
23 fvprc 5714 . . 3  |-  ( -.  B  e.  _V  ->  (
topGen `  B )  =  (/) )
2422, 23eqtr4d 2470 . 2  |-  ( -.  B  e.  _V  ->  (
topGen `  ( B  \  { (/) } ) )  =  ( topGen `  B
) )
2511, 24pm2.61i 158 1  |-  ( topGen `  ( B  \  { (/)
} ) )  =  ( topGen `  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   U.cuni 4007   ` cfv 5446   topGenctg 13657
This theorem is referenced by:  prdsxmslem2  18551
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-topgen 13659
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