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Theorem uffixsn 21949
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 21928 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2 filn0 21886 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
3 intssuni 4634 . . . . . . . 8 (𝐹 ≠ ∅ → 𝐹 𝐹)
41, 2, 33syl 18 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
5 filunibas 21905 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
61, 5syl 17 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
74, 6sseqtrd 3790 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
87sselda 3752 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
98snssd 4476 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ⊆ 𝑋)
10 snex 5037 . . . . 5 {𝐴} ∈ V
1110elpw 4304 . . . 4 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
129, 11sylibr 224 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝒫 𝑋)
13 snidg 4346 . . . 4 (𝐴 𝐹𝐴 ∈ {𝐴})
1413adantl 467 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴 ∈ {𝐴})
15 eleq2 2839 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
1615elrab 3515 . . 3 ({𝐴} ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ ({𝐴} ∈ 𝒫 𝑋𝐴 ∈ {𝐴}))
1712, 14, 16sylanbrc 572 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
18 uffixfr 21947 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
1918biimpa 462 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
2017, 19eleqtrrd 2853 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  {crab 3065  wss 3723  c0 4063  𝒫 cpw 4298  {csn 4317   cuni 4575   cint 4612  cfv 6030  Filcfil 21869  UFilcufil 21923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-fbas 19958  df-fg 19959  df-fil 21870  df-ufil 21925
This theorem is referenced by:  ufildom1  21950  cfinufil  21952  fin1aufil  21956
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