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Theorem snfil 17849
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 3789 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2 eqimss 3360 . . . . 5  |-  ( x  =  A  ->  x  C_  A )
32pm4.71ri 615 . . . 4  |-  ( x  =  A  <->  ( x  C_  A  /\  x  =  A ) )
41, 3bitri 241 . . 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 2924 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
76adantr 452 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  e.  _V )
8 eqid 2404 . . . 4  |-  A  =  A
9 eqsbc3 3160 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
108, 9mpbiri 225 . . 3  |-  ( A  e.  B  ->  [. A  /  x ]. x  =  A )
1110adantr 452 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  [. A  /  x ]. x  =  A )
12 simpr 448 . . . . 5  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  =/=  (/) )
1312necomd 2650 . . . 4  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  (/)  =/=  A
)
1413neneqd 2583 . . 3  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  (/)  =  A )
15 0ex 4299 . . . 4  |-  (/)  e.  _V
16 eqsbc3 3160 . . . 4  |-  ( (/)  e.  _V  ->  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A ) )
1715, 16ax-mp 8 . . 3  |-  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A )
1814, 17sylnibr 297 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  [. (/)  /  x ]. x  =  A )
19 sseq1 3329 . . . . . . 7  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
2019anbi2d 685 . . . . . 6  |-  ( x  =  A  ->  (
( y  C_  A  /\  x  C_  y )  <-> 
( y  C_  A  /\  A  C_  y ) ) )
21 eqss 3323 . . . . . . 7  |-  ( y  =  A  <->  ( y  C_  A  /\  A  C_  y ) )
2221biimpri 198 . . . . . 6  |-  ( ( y  C_  A  /\  A  C_  y )  -> 
y  =  A )
2320, 22syl6bi 220 . . . . 5  |-  ( x  =  A  ->  (
( y  C_  A  /\  x  C_  y )  ->  y  =  A ) )
2423com12 29 . . . 4  |-  ( ( y  C_  A  /\  x  C_  y )  -> 
( x  =  A  ->  y  =  A ) )
25243adant1 975 . . 3  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  y )  ->  (
x  =  A  -> 
y  =  A ) )
26 sbcid 3137 . . 3  |-  ( [. x  /  x ]. x  =  A  <->  x  =  A
)
27 vex 2919 . . . 4  |-  y  e. 
_V
28 eqsbc3 3160 . . . 4  |-  ( y  e.  _V  ->  ( [. y  /  x ]. x  =  A  <->  y  =  A ) )
2927, 28ax-mp 8 . . 3  |-  ( [. y  /  x ]. x  =  A  <->  y  =  A )
3025, 26, 293imtr4g 262 . 2  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  y )  ->  ( [. x  /  x ]. x  =  A  ->  [. y  /  x ]. x  =  A
) )
31 ineq12 3497 . . . . . 6  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  ( A  i^i  A ) )
32 inidm 3510 . . . . . 6  |-  ( A  i^i  A )  =  A
3331, 32syl6eq 2452 . . . . 5  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  A )
3429, 26, 33syl2anb 466 . . . 4  |-  ( (
[. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  ( y  i^i  x )  =  A )
3527inex1 4304 . . . . 5  |-  ( y  i^i  x )  e. 
_V
36 eqsbc3 3160 . . . . 5  |-  ( ( y  i^i  x )  e.  _V  ->  ( [. ( y  i^i  x
)  /  x ]. x  =  A  <->  ( y  i^i  x )  =  A ) )
3735, 36ax-mp 8 . . . 4  |-  ( [. ( y  i^i  x
)  /  x ]. x  =  A  <->  ( y  i^i  x )  =  A )
3834, 37sylibr 204 . . 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 17843 1  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   [.wsbc 3121    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   ` cfv 5413   Filcfil 17830
This theorem is referenced by:  snfbas  17851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-fbas 16654  df-fil 17831
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