Theorem uffixsn 23810

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.

Step	Hyp	Ref	Expression
1		eleq2 2817	. . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
2		ufilfil 23789	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3		filn0 23747	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
4		intssuni 4920	. . . . . . . 8 ⊢ (𝐹 ≠ ∅ → ∩ 𝐹 ⊆ ∪ 𝐹)
5	2, 3, 4	3syl 18	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ ∪ 𝐹)
6		filunibas 23766	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
7	2, 6	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
8	5, 7	sseqtrd 3972	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ 𝑋)
9	8	sselda 3935	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ 𝑋)
10	9	snssd 4760	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝐴} ⊆ 𝑋)
11		snex 5375	. . . . 5 ⊢ {𝐴} ∈ V
12	11	elpw 4555	. . . 4 ⊢ ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
13	10, 12	sylibr 234	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝐴} ∈ 𝒫 𝑋)
14		snidg 4612	. . . 4 ⊢ (𝐴 ∈ ∩ 𝐹 → 𝐴 ∈ {𝐴})
15	14	adantl 481	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ {𝐴})
16	1, 13, 15	elrabd 3650	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝐴} ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
17		uffixfr 23808	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐴 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
18	17	biimpa 476	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
19	16, 18	eleqtrrd 2831	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝐴} ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3394   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ∪ cuni 4858  ∩ cint 4896  ‘cfv 6482  Filcfil 23730  UFilcufil 23784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-fbas 21258  df-fg 21259  df-fil 23731  df-ufil 23786

This theorem is referenced by:  ufildom1  23811  cfinufil  23813  fin1aufil  23817