Theorem uffixsn 24082

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 2852	. . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
2		ufilfil 24061	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3		filn0 24019	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
4		intssuni 4935	. . . . . . . 8 ⊢ (𝐹 ≠ ∅ → ∩ 𝐹 ⊆ ∪ 𝐹)
5	2, 3, 4	3syl 19	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ ∪ 𝐹)
6		filunibas 24038	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
7	2, 6	syl 18	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
8	5, 7	sseqtrd 3973	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ 𝑋)
9	8	sselda 3937	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ 𝑋)
10	9	snssd 4752	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝐴} ⊆ 𝑋)
11		snex 5410	. . . . 5 ⊢ {𝐴} ∈ V
12	11	elpw 4566	. . . 4 ⊢ ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
13	10, 12	sylibr 237	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝐴} ∈ 𝒫 𝑋)
14		snidg 4626	. . . 4 ⊢ (𝐴 ∈ ∩ 𝐹 → 𝐴 ∈ {𝐴})
15	14	adantl 486	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ {𝐴})
16	1, 13, 15	elrabd 3652	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝐴} ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
17		uffixfr 24080	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐴 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
18	17	biimpa 481	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
19	16, 18	eleqtrrd 2866	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝐴} ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  {crab 3416   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872  ∩ cint 4912  ‘cfv 6536  Filcfil 24002  UFilcufil 24056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-fbas 21519  df-fg 21520  df-fil 24003  df-ufil 24058

This theorem is referenced by:  ufildom1  24083  cfinufil  24085  fin1aufil  24089