Theorem uffixsn 21729

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 e. ( UFil ` X ) /\ A e. |^| F ) -> { A } e. F )

Proof of Theorem uffixsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ufilfil 21708	. . . . . . . 8 |- ( F e. ( UFil ` X ) -> F e. ( Fil ` X ) )
2		filn0 21666	. . . . . . . 8 |- ( F e. ( Fil ` X ) -> F =/= (/) )
3		intssuni 4499	. . . . . . . 8 |- ( F =/= (/) -> |^| F C_ U. F )
4	1, 2, 3	3syl 18	. . . . . . 7 |- ( F e. ( UFil ` X ) -> |^| F C_ U. F )
5		filunibas 21685	. . . . . . . 8 |- ( F e. ( Fil ` X ) -> U. F = X )
6	1, 5	syl 17	. . . . . . 7 |- ( F e. ( UFil ` X ) -> U. F = X )
7	4, 6	sseqtrd 3641	. . . . . 6 |- ( F e. ( UFil ` X ) -> |^| F C_ X )
8	7	sselda 3603	. . . . 5 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> A e. X )
9	8	snssd 4340	. . . 4 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> { A } C_ X )
10		snex 4908	. . . . 5 |- { A } e. _V
11	10	elpw 4164	. . . 4 |- ( { A } e. ~P X <-> { A } C_ X )
12	9, 11	sylibr 224	. . 3 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> { A } e. ~P X )
13		snidg 4206	. . . 4 |- ( A e. |^| F -> A e. { A } )
14	13	adantl 482	. . 3 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> A e. { A } )
15		eleq2 2690	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
16	15	elrab 3363	. . 3 |- ( { A } e. { x e. ~P X | A e. x } <-> ( { A } e. ~P X /\ A e. { A } ) )
17	12, 14, 16	sylanbrc 698	. 2 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> { A } e. { x e. ~P X | A e. x } )
18		uffixfr 21727	. . 3 |- ( F e. ( UFil ` X ) -> ( A e. |^| F <-> F = { x e. ~P X | A e. x } ) )
19	18	biimpa 501	. 2 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> F = { x e. ~P X | A e. x } )
20	17, 19	eleqtrrd 2704	1 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> { A } e. F )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |^|cint 4475   ` cfv 5888   Filcfil 21649   UFilcufil 21703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744  df-fil 21650  df-ufil 21705

This theorem is referenced by:  ufildom1  21730  cfinufil  21732  fin1aufil  21736