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Theorem snfil 19562
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 3992 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2 eqimss 3509 . . . . 5  |-  ( x  =  A  ->  x  C_  A )
32pm4.71ri 633 . . . 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 3080 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
76adantr 465 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  e.  _V )
8 eqid 2451 . . . 4  |-  A  =  A
9 eqsbc3 3327 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  A  <->  A  =  A ) )
108, 9mpbiri 233 . . 3  |-  ( A  e.  B  ->  [. A  /  x ]. x  =  A )
1110adantr 465 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  [. A  /  x ]. x  =  A )
12 simpr 461 . . . . 5  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  A  =/=  (/) )
1312necomd 2719 . . . 4  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  (/)  =/=  A
)
1413neneqd 2651 . . 3  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  (/)  =  A )
15 0ex 4523 . . . 4  |-  (/)  e.  _V
16 eqsbc3 3327 . . . 4  |-  ( (/)  e.  _V  ->  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A ) )
1715, 16ax-mp 5 . . 3  |-  ( [. (/)  /  x ]. x  =  A  <->  (/)  =  A )
1814, 17sylnibr 305 . 2  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  -.  [. (/)  /  x ]. x  =  A )
19 sseq1 3478 . . . . . . 7  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
2019anbi2d 703 . . . . . 6  |-  ( x  =  A  ->  (
( y  C_  A  /\  x  C_  y )  <-> 
( y  C_  A  /\  A  C_  y ) ) )
21 eqss 3472 . . . . . . 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 1006 . . 3  |-  ( ( ( A  e.  B  /\  A  =/=  (/) )  /\  y  C_  A  /\  x  C_  y )  ->  (
x  =  A  -> 
y  =  A ) )
26 sbcid 3304 . . 3  |-  ( [. x  /  x ]. x  =  A  <->  x  =  A
)
27 vex 3074 . . . 4  |-  y  e. 
_V
28 eqsbc3 3327 . . . 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 3648 . . . . . 6  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  ( A  i^i  A ) )
32 inidm 3660 . . . . . 6  |-  ( A  i^i  A )  =  A
3331, 32syl6eq 2508 . . . . 5  |-  ( ( y  =  A  /\  x  =  A )  ->  ( y  i^i  x
)  =  A )
3429, 26, 33syl2anb 479 . . . 4  |-  ( (
[. y  /  x ]. x  =  A  /\  [. x  /  x ]. x  =  A
)  ->  ( y  i^i  x )  =  A )
3527inex1 4534 . . . . 5  |-  ( y  i^i  x )  e. 
_V
36 eqsbc3 3327 . . . . 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 19556 1  |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071   [.wsbc 3287    i^i cin 3428    C_ wss 3429   (/)c0 3738   {csn 3978   ` cfv 5519   Filcfil 19543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fv 5527  df-fbas 17932  df-fil 19544
This theorem is referenced by:  snfbas  19564
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