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Theorem snfil 22472
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 4566 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eqimss 4009 . . . . 5 (𝑥 = 𝐴𝑥𝐴)
32pm4.71ri 564 . . . 4 (𝑥 = 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
41, 3bitri 278 . . 3 (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴))
54a1i 11 . 2 ((𝐴𝐵𝐴 ≠ ∅) → (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴)))
6 simpl 486 . 2 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴𝐵)
7 eqid 2824 . . . 4 𝐴 = 𝐴
8 eqsbc3 3803 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
97, 8mpbiri 261 . . 3 (𝐴𝐵[𝐴 / 𝑥]𝑥 = 𝐴)
109adantr 484 . 2 ((𝐴𝐵𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
11 simpr 488 . . . . 5 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 ≠ ∅)
1211necomd 3069 . . . 4 ((𝐴𝐵𝐴 ≠ ∅) → ∅ ≠ 𝐴)
1312neneqd 3019 . . 3 ((𝐴𝐵𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
14 0ex 5197 . . . 4 ∅ ∈ V
15 eqsbc3 3803 . . . 4 (∅ ∈ V → ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴))
1614, 15ax-mp 5 . . 3 ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴)
1713, 16sylnibr 332 . 2 ((𝐴𝐵𝐴 ≠ ∅) → ¬ [∅ / 𝑥]𝑥 = 𝐴)
18 sseq1 3978 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
1918anbi2d 631 . . . . . 6 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) ↔ (𝑦𝐴𝐴𝑦)))
20 eqss 3968 . . . . . . 7 (𝑦 = 𝐴 ↔ (𝑦𝐴𝐴𝑦))
2120biimpri 231 . . . . . 6 ((𝑦𝐴𝐴𝑦) → 𝑦 = 𝐴)
2219, 21syl6bi 256 . . . . 5 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) → 𝑦 = 𝐴))
2322com12 32 . . . 4 ((𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
24233adant1 1127 . . 3 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
25 sbcid 3775 . . 3 ([𝑥 / 𝑥]𝑥 = 𝐴𝑥 = 𝐴)
26 eqsbc3 3803 . . . 4 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴))
2726elv 3485 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
2824, 25, 273imtr4g 299 . 2 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴[𝑦 / 𝑥]𝑥 = 𝐴))
29 ineq12 4169 . . . . . 6 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = (𝐴𝐴))
30 inidm 4180 . . . . . 6 (𝐴𝐴) = 𝐴
3129, 30syl6eq 2875 . . . . 5 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
3227, 25, 31syl2anb 600 . . . 4 (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
33 vex 3483 . . . . . 6 𝑦 ∈ V
3433inex1 5207 . . . . 5 (𝑦𝑥) ∈ V
35 eqsbc3 3803 . . . . 5 ((𝑦𝑥) ∈ V → ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴))
3634, 35ax-mp 5 . . . 4 ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴)
3732, 36sylibr 237 . . 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 22466 1 ((𝐴𝐵𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  Vcvv 3480  [wsbc 3758  cin 3918  wss 3919  c0 4276  {csn 4550  cfv 6343  Filcfil 22453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fv 6351  df-fbas 20542  df-fil 22454
This theorem is referenced by:  snfbas  22474
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