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Theorem snfil 21887
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 4330 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eqimss 3804 . . . . 5 (𝑥 = 𝐴𝑥𝐴)
32pm4.71ri 542 . . . 4 (𝑥 = 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
41, 3bitri 264 . . 3 (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴))
54a1i 11 . 2 ((𝐴𝐵𝐴 ≠ ∅) → (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴)))
6 elex 3361 . . 3 (𝐴𝐵𝐴 ∈ V)
76adantr 466 . 2 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 ∈ V)
8 eqid 2770 . . . 4 𝐴 = 𝐴
9 eqsbc3 3625 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
108, 9mpbiri 248 . . 3 (𝐴𝐵[𝐴 / 𝑥]𝑥 = 𝐴)
1110adantr 466 . 2 ((𝐴𝐵𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
12 simpr 471 . . . . 5 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 ≠ ∅)
1312necomd 2997 . . . 4 ((𝐴𝐵𝐴 ≠ ∅) → ∅ ≠ 𝐴)
1413neneqd 2947 . . 3 ((𝐴𝐵𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
15 0ex 4921 . . . 4 ∅ ∈ V
16 eqsbc3 3625 . . . 4 (∅ ∈ V → ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴))
1715, 16ax-mp 5 . . 3 ([∅ / 𝑥]𝑥 = 𝐴 ↔ ∅ = 𝐴)
1814, 17sylnibr 318 . 2 ((𝐴𝐵𝐴 ≠ ∅) → ¬ [∅ / 𝑥]𝑥 = 𝐴)
19 sseq1 3773 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
2019anbi2d 606 . . . . . 6 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) ↔ (𝑦𝐴𝐴𝑦)))
21 eqss 3765 . . . . . . 7 (𝑦 = 𝐴 ↔ (𝑦𝐴𝐴𝑦))
2221biimpri 218 . . . . . 6 ((𝑦𝐴𝐴𝑦) → 𝑦 = 𝐴)
2320, 22syl6bi 243 . . . . 5 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) → 𝑦 = 𝐴))
2423com12 32 . . . 4 ((𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
25243adant1 1123 . . 3 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
26 sbcid 3602 . . 3 ([𝑥 / 𝑥]𝑥 = 𝐴𝑥 = 𝐴)
27 vex 3352 . . . 4 𝑦 ∈ V
28 eqsbc3 3625 . . . 4 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴))
2927, 28ax-mp 5 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
3025, 26, 293imtr4g 285 . 2 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴[𝑦 / 𝑥]𝑥 = 𝐴))
31 ineq12 3958 . . . . . 6 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = (𝐴𝐴))
32 inidm 3969 . . . . . 6 (𝐴𝐴) = 𝐴
3331, 32syl6eq 2820 . . . . 5 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
3429, 26, 33syl2anb 577 . . . 4 (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
3527inex1 4930 . . . . 5 (𝑦𝑥) ∈ V
36 eqsbc3 3625 . . . . 5 ((𝑦𝑥) ∈ V → ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴))
3735, 36ax-mp 5 . . . 4 ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴)
3834, 37sylibr 224 . . 3 (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → [(𝑦𝑥) / 𝑥]𝑥 = 𝐴)
3938a1i 11 . 2 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝐴) → (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → [(𝑦𝑥) / 𝑥]𝑥 = 𝐴))
405, 7, 11, 18, 30, 39isfild 21881 1 ((𝐴𝐵𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  Vcvv 3349  [wsbc 3585  cin 3720  wss 3721  c0 4061  {csn 4314  cfv 6031  Filcfil 21868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-fbas 19957  df-fil 21869
This theorem is referenced by:  snfbas  21889
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