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Theorem snfil 23888
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 4647 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eqimss 4054 . . . . 5 (𝑥 = 𝐴𝑥𝐴)
32pm4.71ri 560 . . . 4 (𝑥 = 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
41, 3bitri 275 . . 3 (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴))
54a1i 11 . 2 ((𝐴𝐵𝐴 ≠ ∅) → (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴)))
6 simpl 482 . 2 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴𝐵)
7 eqid 2735 . . . 4 𝐴 = 𝐴
8 eqsbc1 3841 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
97, 8mpbiri 258 . . 3 (𝐴𝐵[𝐴 / 𝑥]𝑥 = 𝐴)
109adantr 480 . 2 ((𝐴𝐵𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
11 simpr 484 . . . . 5 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 ≠ ∅)
1211necomd 2994 . . . 4 ((𝐴𝐵𝐴 ≠ ∅) → ∅ ≠ 𝐴)
1312neneqd 2943 . . 3 ((𝐴𝐵𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
14 0ex 5313 . . . 4 ∅ ∈ V
15 eqsbc1 3841 . . . 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 4021 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
1918anbi2d 630 . . . . . 6 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) ↔ (𝑦𝐴𝐴𝑦)))
20 eqss 4011 . . . . . . 7 (𝑦 = 𝐴 ↔ (𝑦𝐴𝐴𝑦))
2120biimpri 228 . . . . . 6 ((𝑦𝐴𝐴𝑦) → 𝑦 = 𝐴)
2219, 21biimtrdi 253 . . . . 5 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) → 𝑦 = 𝐴))
2322com12 32 . . . 4 ((𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
24233adant1 1129 . . 3 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
25 sbcid 3808 . . 3 ([𝑥 / 𝑥]𝑥 = 𝐴𝑥 = 𝐴)
26 eqsbc1 3841 . . . 4 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴))
2726elv 3483 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
2824, 25, 273imtr4g 296 . 2 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴[𝑦 / 𝑥]𝑥 = 𝐴))
29 ineq12 4223 . . . . . 6 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = (𝐴𝐴))
30 inidm 4235 . . . . . 6 (𝐴𝐴) = 𝐴
3129, 30eqtrdi 2791 . . . . 5 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
3227, 25, 31syl2anb 598 . . . 4 (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
33 vex 3482 . . . . . 6 𝑦 ∈ V
3433inex1 5323 . . . . 5 (𝑦𝑥) ∈ V
35 eqsbc1 3841 . . . . 5 ((𝑦𝑥) ∈ V → ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴))
3634, 35ax-mp 5 . . . 4 ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴)
3732, 36sylibr 234 . . 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 23882 1 ((𝐴𝐵𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  [wsbc 3791  cin 3962  wss 3963  c0 4339  {csn 4631  cfv 6563  Filcfil 23869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-fbas 21379  df-fil 23870
This theorem is referenced by:  snfbas  23890
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