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

Theorem elnei 19478
Description: A point belongs to any of its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)
Assertion
Ref Expression
elnei  |-  ( ( J  e.  Top  /\  P  e.  A  /\  N  e.  ( ( nei `  J ) `  { P } ) )  ->  P  e.  N
)

Proof of Theorem elnei
StepHypRef Expression
1 ssnei 19477 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  { P } ) )  ->  { P }  C_  N )
213adant2 1014 . 2  |-  ( ( J  e.  Top  /\  P  e.  A  /\  N  e.  ( ( nei `  J ) `  { P } ) )  ->  { P }  C_  N )
3 snssg 4144 . . 3  |-  ( P  e.  A  ->  ( P  e.  N  <->  { P }  C_  N ) )
433ad2ant2 1017 . 2  |-  ( ( J  e.  Top  /\  P  e.  A  /\  N  e.  ( ( nei `  J ) `  { P } ) )  ->  ( P  e.  N  <->  { P }  C_  N ) )
52, 4mpbird 232 1  |-  ( ( J  e.  Top  /\  P  e.  A  /\  N  e.  ( ( nei `  J ) `  { P } ) )  ->  P  e.  N
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 972    e. wcel 1802    C_ wss 3458   {csn 4010   ` cfv 5574   Topctop 19261   neicnei 19464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-top 19266  df-nei 19465
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator