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Theorem snfil 20534
Description: A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
snfil  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )

Proof of Theorem snfil
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsn 4030 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2 eqimss 3541 . . . . 5  |-  ( x  =  A  ->  x  C_  A )
32pm4.71ri 631 . . . 4  |-  ( x  =  A  <->  ( x  C_  A  /\  x  =  A ) )
41, 3bitri 249 . . 3  |-  ( x  e.  { A }  <->  ( x  C_  A  /\  x  =  A )
)
54a1i 11 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  (
x  e.  { A } 
<->  ( x  C_  A  /\  x  =  A
) ) )
6 elex 3115 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
76adantr 463 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  e.  _V )
8 eqid 2454 . . . 4  |-  A  =  A
9 eqsbc3 3364 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
108, 9mpbiri 233 . . 3  |-  ( A  e.  B  ->  [. A  /  x ]. x  =  A )
1110adantr 463 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  [. A  /  x ]. x  =  A )
12 simpr 459 . . . . 5  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  =/=  (/) )
1312necomd 2725 . . . 4  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  (/)  =/=  A
)
1413neneqd 2656 . . 3  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  (/)  =  A )
15 0ex 4569 . . . 4  |-  (/)  e.  _V
16 eqsbc3 3364 . . . 4  |-  ( (/)  e.  _V  ->  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A ) )
1715, 16ax-mp 5 . . 3  |-  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A )
1814, 17sylnibr 303 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  [. (/)  /  x ]. x  =  A )
19 sseq1 3510 . . . . . . 7  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
2019anbi2d 701 . . . . . 6  |-  ( x  =  A  ->  (
( y  C_  A  /\  x  C_  y )  <-> 
( y  C_  A  /\  A  C_  y ) ) )
21 eqss 3504 . . . . . . 7  |-  ( y  =  A  <->  ( y  C_  A  /\  A  C_  y ) )
2221biimpri 206 . . . . . 6  |-  ( ( y  C_  A  /\  A  C_  y )  -> 
y  =  A )
2320, 22syl6bi 228 . . . . 5  |-  ( x  =  A  ->  (
( y  C_  A  /\  x  C_  y )  ->  y  =  A ) )
2423com12 31 . . . 4  |-  ( ( y  C_  A  /\  x  C_  y )  -> 
( x  =  A  ->  y  =  A ) )
25243adant1 1012 . . 3  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  y )  ->  (
x  =  A  -> 
y  =  A ) )
26 sbcid 3341 . . 3  |-  ( [. x  /  x ]. x  =  A  <->  x  =  A
)
27 vex 3109 . . . 4  |-  y  e. 
_V
28 eqsbc3 3364 . . . 4  |-  ( y  e.  _V  ->  ( [. y  /  x ]. x  =  A  <->  y  =  A ) )
2927, 28ax-mp 5 . . 3  |-  ( [. y  /  x ]. x  =  A  <->  y  =  A )
3025, 26, 293imtr4g 270 . 2  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  y )  ->  ( [. x  /  x ]. x  =  A  ->  [. y  /  x ]. x  =  A
) )
31 ineq12 3681 . . . . . 6  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  ( A  i^i  A ) )
32 inidm 3693 . . . . . 6  |-  ( A  i^i  A )  =  A
3331, 32syl6eq 2511 . . . . 5  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  A )
3429, 26, 33syl2anb 477 . . . 4  |-  ( (
[. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  ( y  i^i  x )  =  A )
3527inex1 4578 . . . . 5  |-  ( y  i^i  x )  e. 
_V
36 eqsbc3 3364 . . . . 5  |-  ( ( y  i^i  x )  e.  _V  ->  ( [. ( y  i^i  x
)  /  x ]. x  =  A  <->  ( y  i^i  x )  =  A ) )
3735, 36ax-mp 5 . . . 4  |-  ( [. ( y  i^i  x
)  /  x ]. x  =  A  <->  ( y  i^i  x )  =  A )
3834, 37sylibr 212 . . 3  |-  ( (
[. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  [. ( y  i^i  x )  /  x ]. x  =  A )
3938a1i 11 . 2  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  A )  ->  (
( [. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  [. ( y  i^i  x )  /  x ]. x  =  A ) )
405, 7, 11, 18, 30, 39isfild 20528 1  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   [.wsbc 3324    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   ` cfv 5570   Filcfil 20515
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
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-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  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-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-fbas 18614  df-fil 20516
This theorem is referenced by:  snfbas  20536
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