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Theorem snfil 23015
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 ((𝐴𝐵𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))

Proof of Theorem snfil
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 velsn 4577 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eqimss 3977 . . . . 5 (𝑥 = 𝐴𝑥𝐴)
32pm4.71ri 561 . . . 4 (𝑥 = 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
41, 3bitri 274 . . 3 (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴))
54a1i 11 . 2 ((𝐴𝐵𝐴 ≠ ∅) → (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴)))
6 simpl 483 . 2 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴𝐵)
7 eqid 2738 . . . 4 𝐴 = 𝐴
8 eqsbc1 3765 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
97, 8mpbiri 257 . . 3 (𝐴𝐵[𝐴 / 𝑥]𝑥 = 𝐴)
109adantr 481 . 2 ((𝐴𝐵𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
11 simpr 485 . . . . 5 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 ≠ ∅)
1211necomd 2999 . . . 4 ((𝐴𝐵𝐴 ≠ ∅) → ∅ ≠ 𝐴)
1312neneqd 2948 . . 3 ((𝐴𝐵𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
14 0ex 5231 . . . 4 ∅ ∈ V
15 eqsbc1 3765 . . . 4 (∅ ∈ V → ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴))
1614, 15ax-mp 5 . . 3 ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴)
1713, 16sylnibr 329 . 2 ((𝐴𝐵𝐴 ≠ ∅) → ¬ [∅ / 𝑥]𝑥 = 𝐴)
18 sseq1 3946 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
1918anbi2d 629 . . . . . 6 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) ↔ (𝑦𝐴𝐴𝑦)))
20 eqss 3936 . . . . . . 7 (𝑦 = 𝐴 ↔ (𝑦𝐴𝐴𝑦))
2120biimpri 227 . . . . . 6 ((𝑦𝐴𝐴𝑦) → 𝑦 = 𝐴)
2219, 21syl6bi 252 . . . . 5 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) → 𝑦 = 𝐴))
2322com12 32 . . . 4 ((𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
24233adant1 1129 . . 3 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
25 sbcid 3733 . . 3 ([𝑥 / 𝑥]𝑥 = 𝐴𝑥 = 𝐴)
26 eqsbc1 3765 . . . 4 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴))
2726elv 3438 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
2824, 25, 273imtr4g 296 . 2 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴[𝑦 / 𝑥]𝑥 = 𝐴))
29 ineq12 4141 . . . . . 6 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = (𝐴𝐴))
30 inidm 4152 . . . . . 6 (𝐴𝐴) = 𝐴
3129, 30eqtrdi 2794 . . . . 5 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
3227, 25, 31syl2anb 598 . . . 4 (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
33 vex 3436 . . . . . 6 𝑦 ∈ V
3433inex1 5241 . . . . 5 (𝑦𝑥) ∈ V
35 eqsbc1 3765 . . . . 5 ((𝑦𝑥) ∈ V → ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴))
3634, 35ax-mp 5 . . . 4 ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴)
3732, 36sylibr 233 . . 3 (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → [(𝑦𝑥) / 𝑥]𝑥 = 𝐴)
3837a1i 11 . 2 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝐴) → (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → [(𝑦𝑥) / 𝑥]𝑥 = 𝐴))
395, 6, 10, 17, 28, 38isfild 23009 1 ((𝐴𝐵𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  Vcvv 3432  [wsbc 3716  cin 3886  wss 3887  c0 4256  {csn 4561  cfv 6433  Filcfil 22996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-fbas 20594  df-fil 22997
This theorem is referenced by:  snfbas  23017
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