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Theorem snfil 17553
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 ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem snfil
StepHypRef Expression
1 elsn 3656 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2 eqimss 3231 . . . . 5  |-  ( x  =  A  ->  x  C_  A )
32pm4.71ri 616 . . . 4  |-  ( x  =  A  <->  ( x  C_  A  /\  x  =  A ) )
41, 3bitri 242 . . 3  |-  ( x  e.  { A }  <->  ( x  C_  A  /\  x  =  A )
)
54a1i 12 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  (
x  e.  { A } 
<->  ( x  C_  A  /\  x  =  A
) ) )
6 elex 2797 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
76adantr 453 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  e.  _V )
8 eqid 2284 . . . 4  |-  A  =  A
9 eqsbc3 3031 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
108, 9mpbiri 226 . . 3  |-  ( A  e.  B  ->  [. A  /  x ]. x  =  A )
1110adantr 453 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  [. A  /  x ]. x  =  A )
12 simpr 449 . . . . 5  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  =/=  (/) )
1312necomd 2530 . . . 4  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  (/)  =/=  A
)
1413neneqd 2463 . . 3  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  (/)  =  A )
15 0ex 4151 . . . 4  |-  (/)  e.  _V
16 eqsbc3 3031 . . . 4  |-  ( (/)  e.  _V  ->  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A ) )
1715, 16ax-mp 10 . . 3  |-  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A )
1814, 17sylnibr 298 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  [. (/)  /  x ]. x  =  A )
19 sseq1 3200 . . . . . . 7  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
2019anbi2d 686 . . . . . 6  |-  ( x  =  A  ->  (
( y  C_  A  /\  x  C_  y )  <-> 
( y  C_  A  /\  A  C_  y ) ) )
21 eqss 3195 . . . . . . 7  |-  ( y  =  A  <->  ( y  C_  A  /\  A  C_  y ) )
2221biimpri 199 . . . . . 6  |-  ( ( y  C_  A  /\  A  C_  y )  -> 
y  =  A )
2320, 22syl6bi 221 . . . . 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 vex 2792 . . . 4  |-  x  e. 
_V
27 eqsbc3 3031 . . . 4  |-  ( x  e.  _V  ->  ( [. x  /  x ]. x  =  A  <->  x  =  A ) )
2826, 27ax-mp 10 . . 3  |-  ( [. x  /  x ]. x  =  A  <->  x  =  A
)
29 vex 2792 . . . 4  |-  y  e. 
_V
30 eqsbc3 3031 . . . 4  |-  ( y  e.  _V  ->  ( [. y  /  x ]. x  =  A  <->  y  =  A ) )
3129, 30ax-mp 10 . . 3  |-  ( [. y  /  x ]. x  =  A  <->  y  =  A )
3225, 28, 313imtr4g 263 . 2  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  y )  ->  ( [. x  /  x ]. x  =  A  ->  [. y  /  x ]. x  =  A
) )
33 ineq12 3366 . . . . . 6  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  ( A  i^i  A ) )
34 inidm 3379 . . . . . 6  |-  ( A  i^i  A )  =  A
3533, 34syl6eq 2332 . . . . 5  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  A )
3631, 28, 35syl2anb 467 . . . 4  |-  ( (
[. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  ( y  i^i  x )  =  A )
3729inex1 4156 . . . . 5  |-  ( y  i^i  x )  e. 
_V
38 eqsbc3 3031 . . . . 5  |-  ( ( y  i^i  x )  e.  _V  ->  ( [. ( y  i^i  x
)  /  x ]. x  =  A  <->  ( y  i^i  x )  =  A ) )
3937, 38ax-mp 10 . . . 4  |-  ( [. ( y  i^i  x
)  /  x ]. x  =  A  <->  ( y  i^i  x )  =  A )
4036, 39sylibr 205 . . 3  |-  ( (
[. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  [. ( y  i^i  x )  /  x ]. x  =  A )
4140a1i 12 . 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 17547 1  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   _Vcvv 2789   [.wsbc 2992    i^i cin 3152    C_ wss 3153   (/)c0 3456   {csn 3641   ` cfv 5221   Filcfil 17534
This theorem is referenced by:  snfbas  17555  efilcp  24951
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-fbas 17514  df-fil 17535
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