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Theorem uffixsn 22449
Description: The singleton of the generator of a fixed ultrafilter is in the filter. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
uffixsn ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝐹)

Proof of Theorem uffixsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2905 . . 3 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
2 ufilfil 22428 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filn0 22386 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
4 intssuni 4895 . . . . . . . 8 (𝐹 ≠ ∅ → 𝐹 𝐹)
52, 3, 43syl 18 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
6 filunibas 22405 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
72, 6syl 17 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
85, 7sseqtrd 4010 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
98sselda 3970 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
109snssd 4740 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ⊆ 𝑋)
11 snex 5327 . . . . 5 {𝐴} ∈ V
1211elpw 4548 . . . 4 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
1310, 12sylibr 235 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝒫 𝑋)
14 snidg 4595 . . . 4 (𝐴 𝐹𝐴 ∈ {𝐴})
1514adantl 482 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴 ∈ {𝐴})
161, 13, 15elrabd 3685 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
17 uffixfr 22447 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
1817biimpa 477 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
1916, 18eleqtrrd 2920 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wne 3020  {crab 3146  wss 3939  c0 4294  𝒫 cpw 4541  {csn 4563   cuni 4836   cint 4873  cfv 6351  Filcfil 22369  UFilcufil 22423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-fbas 20458  df-fg 20459  df-fil 22370  df-ufil 22425
This theorem is referenced by:  ufildom1  22450  cfinufil  22452  fin1aufil  22456
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