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Theorem tg1 17021
Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
Assertion
Ref Expression
tg1  |-  ( A  e.  ( topGen `  B
)  ->  A  C_  U. B
)

Proof of Theorem tg1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5749 . 2  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
2 eltg2 17015 . . 3  |-  ( B  e.  dom  topGen  ->  ( A  e.  ( topGen `  B )  <->  ( A  C_ 
U. B  /\  A. x  e.  A  E. y  e.  B  (
x  e.  y  /\  y  C_  A ) ) ) )
32simprbda 607 . 2  |-  ( ( B  e.  dom  topGen  /\  A  e.  ( topGen `  B )
)  ->  A  C_  U. B
)
41, 3mpancom 651 1  |-  ( A  e.  ( topGen `  B
)  ->  A  C_  U. B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   U.cuni 4007   dom cdm 4870   ` cfv 5446   topGenctg 13657
This theorem is referenced by:  tgcl  17026  ontgval  26173
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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