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Theorem snfil 17559
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 3655 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2 eqimss 3230 . . . . 5  |-  ( x  =  A  ->  x  C_  A )
32pm4.71ri 614 . . . 4  |-  ( x  =  A  <->  ( x  C_  A  /\  x  =  A ) )
41, 3bitri 240 . . 3  |-  ( x  e.  { A }  <->  ( x  C_  A  /\  x  =  A )
)
54a1i 10 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  (
x  e.  { A } 
<->  ( x  C_  A  /\  x  =  A
) ) )
6 elex 2796 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
76adantr 451 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  e.  _V )
8 eqid 2283 . . . 4  |-  A  =  A
9 eqsbc3 3030 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
108, 9mpbiri 224 . . 3  |-  ( A  e.  B  ->  [. A  /  x ]. x  =  A )
1110adantr 451 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  [. A  /  x ]. x  =  A )
12 simpr 447 . . . . 5  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  =/=  (/) )
1312necomd 2529 . . . 4  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  (/)  =/=  A
)
1413neneqd 2462 . . 3  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  (/)  =  A )
15 0ex 4150 . . . 4  |-  (/)  e.  _V
16 eqsbc3 3030 . . . 4  |-  ( (/)  e.  _V  ->  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A ) )
1715, 16ax-mp 8 . . 3  |-  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A )
1814, 17sylnibr 296 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  [. (/)  /  x ]. x  =  A )
19 sseq1 3199 . . . . . . 7  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
2019anbi2d 684 . . . . . 6  |-  ( x  =  A  ->  (
( y  C_  A  /\  x  C_  y )  <-> 
( y  C_  A  /\  A  C_  y ) ) )
21 eqss 3194 . . . . . . 7  |-  ( y  =  A  <->  ( y  C_  A  /\  A  C_  y ) )
2221biimpri 197 . . . . . 6  |-  ( ( y  C_  A  /\  A  C_  y )  -> 
y  =  A )
2320, 22syl6bi 219 . . . . 5  |-  ( x  =  A  ->  (
( y  C_  A  /\  x  C_  y )  ->  y  =  A ) )
2423com12 27 . . . 4  |-  ( ( y  C_  A  /\  x  C_  y )  -> 
( x  =  A  ->  y  =  A ) )
25243adant1 973 . . 3  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  y )  ->  (
x  =  A  -> 
y  =  A ) )
26 vex 2791 . . . 4  |-  x  e. 
_V
27 eqsbc3 3030 . . . 4  |-  ( x  e.  _V  ->  ( [. x  /  x ]. x  =  A  <->  x  =  A ) )
2826, 27ax-mp 8 . . 3  |-  ( [. x  /  x ]. x  =  A  <->  x  =  A
)
29 vex 2791 . . . 4  |-  y  e. 
_V
30 eqsbc3 3030 . . . 4  |-  ( y  e.  _V  ->  ( [. y  /  x ]. x  =  A  <->  y  =  A ) )
3129, 30ax-mp 8 . . 3  |-  ( [. y  /  x ]. x  =  A  <->  y  =  A )
3225, 28, 313imtr4g 261 . 2  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  y )  ->  ( [. x  /  x ]. x  =  A  ->  [. y  /  x ]. x  =  A
) )
33 ineq12 3365 . . . . . 6  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  ( A  i^i  A ) )
34 inidm 3378 . . . . . 6  |-  ( A  i^i  A )  =  A
3533, 34syl6eq 2331 . . . . 5  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  A )
3631, 28, 35syl2anb 465 . . . 4  |-  ( (
[. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  ( y  i^i  x )  =  A )
3729inex1 4155 . . . . 5  |-  ( y  i^i  x )  e. 
_V
38 eqsbc3 3030 . . . . 5  |-  ( ( y  i^i  x )  e.  _V  ->  ( [. ( y  i^i  x
)  /  x ]. x  =  A  <->  ( y  i^i  x )  =  A ) )
3937, 38ax-mp 8 . . . 4  |-  ( [. ( y  i^i  x
)  /  x ]. x  =  A  <->  ( y  i^i  x )  =  A )
4036, 39sylibr 203 . . 3  |-  ( (
[. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  [. ( y  i^i  x )  /  x ]. x  =  A )
4140a1i 10 . 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 ) )
425, 7, 11, 18, 32, 41isfild 17553 1  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   [.wsbc 2991    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   ` cfv 5255   Filcfil 17540
This theorem is referenced by:  snfbas  17561  efilcp  24964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-fbas 17520  df-fil 17541
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