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Theorem uffixsn 22625
 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 2840 . . 3 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
2 ufilfil 22604 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filn0 22562 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
4 intssuni 4860 . . . . . . . 8 (𝐹 ≠ ∅ → 𝐹 𝐹)
52, 3, 43syl 18 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
6 filunibas 22581 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
72, 6syl 17 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
85, 7sseqtrd 3932 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
98sselda 3892 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
109snssd 4699 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ⊆ 𝑋)
11 snex 5300 . . . . 5 {𝐴} ∈ V
1211elpw 4498 . . . 4 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
1310, 12sylibr 237 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝒫 𝑋)
14 snidg 4556 . . . 4 (𝐴 𝐹𝐴 ∈ {𝐴})
1514adantl 485 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴 ∈ {𝐴})
161, 13, 15elrabd 3604 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
17 uffixfr 22623 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
1817biimpa 480 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
1916, 18eleqtrrd 2855 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  {crab 3074   ⊆ wss 3858  ∅c0 4225  𝒫 cpw 4494  {csn 4522  ∪ cuni 4798  ∩ cint 4838  ‘cfv 6335  Filcfil 22545  UFilcufil 22599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-fbas 20163  df-fg 20164  df-fil 22546  df-ufil 22601 This theorem is referenced by:  ufildom1  22626  cfinufil  22628  fin1aufil  22632
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