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Theorem uffixsn 23949
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 2828 . . 3 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
2 ufilfil 23928 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filn0 23886 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
4 intssuni 4975 . . . . . . . 8 (𝐹 ≠ ∅ → 𝐹 𝐹)
52, 3, 43syl 18 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
6 filunibas 23905 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
72, 6syl 17 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
85, 7sseqtrd 4036 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
98sselda 3995 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
109snssd 4814 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ⊆ 𝑋)
11 snex 5442 . . . . 5 {𝐴} ∈ V
1211elpw 4609 . . . 4 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
1310, 12sylibr 234 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝒫 𝑋)
14 snidg 4665 . . . 4 (𝐴 𝐹𝐴 ∈ {𝐴})
1514adantl 481 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴 ∈ {𝐴})
161, 13, 15elrabd 3697 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
17 uffixfr 23947 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
1817biimpa 476 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
1916, 18eleqtrrd 2842 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  {crab 3433  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912   cint 4951  cfv 6563  Filcfil 23869  UFilcufil 23923
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-int 4952  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-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380  df-fil 23870  df-ufil 23925
This theorem is referenced by:  ufildom1  23950  cfinufil  23952  fin1aufil  23956
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