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Theorem snfil 21478
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 4141 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eqimss 3620 . . . . 5 (𝑥 = 𝐴𝑥𝐴)
32pm4.71ri 663 . . . 4 (𝑥 = 𝐴 ↔ (𝑥𝐴𝑥 = 𝐴))
41, 3bitri 263 . . 3 (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴))
54a1i 11 . 2 ((𝐴𝐵𝐴 ≠ ∅) → (𝑥 ∈ {𝐴} ↔ (𝑥𝐴𝑥 = 𝐴)))
6 elex 3185 . . 3 (𝐴𝐵𝐴 ∈ V)
76adantr 480 . 2 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 ∈ V)
8 eqid 2610 . . . 4 𝐴 = 𝐴
9 eqsbc3 3442 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
108, 9mpbiri 247 . . 3 (𝐴𝐵[𝐴 / 𝑥]𝑥 = 𝐴)
1110adantr 480 . 2 ((𝐴𝐵𝐴 ≠ ∅) → [𝐴 / 𝑥]𝑥 = 𝐴)
12 simpr 476 . . . . 5 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 ≠ ∅)
1312necomd 2837 . . . 4 ((𝐴𝐵𝐴 ≠ ∅) → ∅ ≠ 𝐴)
1413neneqd 2787 . . 3 ((𝐴𝐵𝐴 ≠ ∅) → ¬ ∅ = 𝐴)
15 0ex 4718 . . . 4 ∅ ∈ V
16 eqsbc3 3442 . . . 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 3589 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
2019anbi2d 736 . . . . . 6 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) ↔ (𝑦𝐴𝐴𝑦)))
21 eqss 3583 . . . . . . 7 (𝑦 = 𝐴 ↔ (𝑦𝐴𝐴𝑦))
2221biimpri 217 . . . . . 6 ((𝑦𝐴𝐴𝑦) → 𝑦 = 𝐴)
2320, 22syl6bi 242 . . . . 5 (𝑥 = 𝐴 → ((𝑦𝐴𝑥𝑦) → 𝑦 = 𝐴))
2423com12 32 . . . 4 ((𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
25243adant1 1072 . . 3 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → (𝑥 = 𝐴𝑦 = 𝐴))
26 sbcid 3419 . . 3 ([𝑥 / 𝑥]𝑥 = 𝐴𝑥 = 𝐴)
27 vex 3176 . . . 4 𝑦 ∈ V
28 eqsbc3 3442 . . . 4 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴))
2927, 28ax-mp 5 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
3025, 26, 293imtr4g 284 . 2 (((𝐴𝐵𝐴 ≠ ∅) ∧ 𝑦𝐴𝑥𝑦) → ([𝑥 / 𝑥]𝑥 = 𝐴[𝑦 / 𝑥]𝑥 = 𝐴))
31 ineq12 3771 . . . . . 6 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = (𝐴𝐴))
32 inidm 3784 . . . . . 6 (𝐴𝐴) = 𝐴
3331, 32syl6eq 2660 . . . . 5 ((𝑦 = 𝐴𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
3429, 26, 33syl2anb 495 . . . 4 (([𝑦 / 𝑥]𝑥 = 𝐴[𝑥 / 𝑥]𝑥 = 𝐴) → (𝑦𝑥) = 𝐴)
3527inex1 4727 . . . . 5 (𝑦𝑥) ∈ V
36 eqsbc3 3442 . . . . 5 ((𝑦𝑥) ∈ V → ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴))
3735, 36ax-mp 5 . . . 4 ([(𝑦𝑥) / 𝑥]𝑥 = 𝐴 ↔ (𝑦𝑥) = 𝐴)
3834, 37sylibr 223 . . 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 21472 1 ((𝐴𝐵𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  [wsbc 3402  cin 3539  wss 3540  c0 3874  {csn 4125  cfv 5804  Filcfil 21459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-fbas 19564  df-fil 21460
This theorem is referenced by:  snfbas  21480
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