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

Theorem tgdom 19994
Description: A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
tgdom  |-  ( B  e.  V  ->  ( topGen `
 B )  ~<_  ~P B )

Proof of Theorem tgdom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4587 . 2  |-  ( B  e.  V  ->  ~P B  e.  _V )
2 inss1 3652 . . . . 5  |-  ( B  i^i  ~P x ) 
C_  B
3 vex 3048 . . . . . . . 8  |-  x  e. 
_V
43pwex 4586 . . . . . . 7  |-  ~P x  e.  _V
54inex2 4545 . . . . . 6  |-  ( B  i^i  ~P x )  e.  _V
65elpw 3957 . . . . 5  |-  ( ( B  i^i  ~P x
)  e.  ~P B  <->  ( B  i^i  ~P x
)  C_  B )
72, 6mpbir 213 . . . 4  |-  ( B  i^i  ~P x )  e.  ~P B
87a1i 11 . . 3  |-  ( x  e.  ( topGen `  B
)  ->  ( B  i^i  ~P x )  e. 
~P B )
9 unieq 4206 . . . . . . 7  |-  ( ( B  i^i  ~P x
)  =  ( B  i^i  ~P y )  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P y
) )
109adantl 468 . . . . . 6  |-  ( ( ( x  e.  (
topGen `  B )  /\  y  e.  ( topGen `  B ) )  /\  ( B  i^i  ~P x
)  =  ( B  i^i  ~P y ) )  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P y ) )
11 eltg4i 19975 . . . . . . 7  |-  ( x  e.  ( topGen `  B
)  ->  x  =  U. ( B  i^i  ~P x ) )
1211ad2antrr 732 . . . . . 6  |-  ( ( ( x  e.  (
topGen `  B )  /\  y  e.  ( topGen `  B ) )  /\  ( B  i^i  ~P x
)  =  ( B  i^i  ~P y ) )  ->  x  =  U. ( B  i^i  ~P x ) )
13 eltg4i 19975 . . . . . . 7  |-  ( y  e.  ( topGen `  B
)  ->  y  =  U. ( B  i^i  ~P y ) )
1413ad2antlr 733 . . . . . 6  |-  ( ( ( x  e.  (
topGen `  B )  /\  y  e.  ( topGen `  B ) )  /\  ( B  i^i  ~P x
)  =  ( B  i^i  ~P y ) )  ->  y  =  U. ( B  i^i  ~P y ) )
1510, 12, 143eqtr4d 2495 . . . . 5  |-  ( ( ( x  e.  (
topGen `  B )  /\  y  e.  ( topGen `  B ) )  /\  ( B  i^i  ~P x
)  =  ( B  i^i  ~P y ) )  ->  x  =  y )
1615ex 436 . . . 4  |-  ( ( x  e.  ( topGen `  B )  /\  y  e.  ( topGen `  B )
)  ->  ( ( B  i^i  ~P x )  =  ( B  i^i  ~P y )  ->  x  =  y ) )
17 pweq 3954 . . . . 5  |-  ( x  =  y  ->  ~P x  =  ~P y
)
1817ineq2d 3634 . . . 4  |-  ( x  =  y  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P y ) )
1916, 18impbid1 207 . . 3  |-  ( ( x  e.  ( topGen `  B )  /\  y  e.  ( topGen `  B )
)  ->  ( ( B  i^i  ~P x )  =  ( B  i^i  ~P y )  <->  x  =  y ) )
208, 19dom2 7612 . 2  |-  ( ~P B  e.  _V  ->  (
topGen `  B )  ~<_  ~P B )
211, 20syl 17 1  |-  ( B  e.  V  ->  ( topGen `
 B )  ~<_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    i^i cin 3403    C_ wss 3404   ~Pcpw 3951   U.cuni 4198   class class class wbr 4402   ` cfv 5582    ~<_ cdom 7567   topGenctg 15336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-dom 7571  df-topgen 15342
This theorem is referenced by:  2ndcredom  20465  kelac2lem  35922
  Copyright terms: Public domain W3C validator