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Theorem uffixsn 22056
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 ufilfil 22035 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2 filn0 21993 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
3 intssuni 4690 . . . . . . . 8 (𝐹 ≠ ∅ → 𝐹 𝐹)
41, 2, 33syl 18 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
5 filunibas 22012 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
61, 5syl 17 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
74, 6sseqtrd 3838 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
87sselda 3799 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
98snssd 4529 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ⊆ 𝑋)
10 snex 5100 . . . . 5 {𝐴} ∈ V
1110elpw 4356 . . . 4 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
129, 11sylibr 226 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝒫 𝑋)
13 snidg 4399 . . . 4 (𝐴 𝐹𝐴 ∈ {𝐴})
1413adantl 474 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴 ∈ {𝐴})
15 eleq2 2868 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
1615elrab 3557 . . 3 ({𝐴} ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ ({𝐴} ∈ 𝒫 𝑋𝐴 ∈ {𝐴}))
1712, 14, 16sylanbrc 579 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
18 uffixfr 22054 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
1918biimpa 469 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
2017, 19eleqtrrd 2882 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2972  {crab 3094  wss 3770  c0 4116  𝒫 cpw 4350  {csn 4369   cuni 4629   cint 4668  cfv 6102  Filcfil 21976  UFilcufil 22030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-int 4669  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-fbas 20064  df-fg 20065  df-fil 21977  df-ufil 22032
This theorem is referenced by:  ufildom1  22057  cfinufil  22059  fin1aufil  22063
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