Theorem snfil 20879

Description: A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
snfil	|- ( ( A e. B /\ A =/= (/) ) -> { A } e. ( Fil ` A ) )

Proof of Theorem snfil

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsn 3982	. . . 4 |- ( x e. { A } <-> x = A )
2		eqimss 3484	. . . . 5 |- ( x = A -> x C_ A )
3	2	pm4.71ri 639	. . . 4 |- ( x = A <-> ( x C_ A /\ x = A ) )
4	1, 3	bitri 253	. . 3 |- ( x e. { A } <-> ( x C_ A /\ x = A ) )
5	4	a1i 11	. 2 |- ( ( A e. B /\ A =/= (/) ) -> ( x e. { A } <-> ( x C_ A /\ x = A ) ) )
6		elex 3054	. . 3 |- ( A e. B -> A e. _V )
7	6	adantr 467	. 2 |- ( ( A e. B /\ A =/= (/) ) -> A e. _V )
8		eqid 2451	. . . 4 |- A = A
9		eqsbc3 3307	. . . 4 |- ( A e. B -> ( [. A / x ]. x = A <-> A = A ) )
10	8, 9	mpbiri 237	. . 3 |- ( A e. B -> [. A / x ]. x = A )
11	10	adantr 467	. 2 |- ( ( A e. B /\ A =/= (/) ) -> [. A / x ]. x = A )
12		simpr 463	. . . . 5 |- ( ( A e. B /\ A =/= (/) ) -> A =/= (/) )
13	12	necomd 2679	. . . 4 |- ( ( A e. B /\ A =/= (/) ) -> (/) =/= A )
14	13	neneqd 2629	. . 3 |- ( ( A e. B /\ A =/= (/) ) -> -. (/) = A )
15		0ex 4535	. . . 4 |- (/) e. _V
16		eqsbc3 3307	. . . 4 |- ( (/) e. _V -> ( [. (/) / x ]. x = A <-> (/) = A ) )
17	15, 16	ax-mp 5	. . 3 |- ( [. (/) / x ]. x = A <-> (/) = A )
18	14, 17	sylnibr 307	. 2 |- ( ( A e. B /\ A =/= (/) ) -> -. [. (/) / x ]. x = A )
19		sseq1 3453	. . . . . . 7 |- ( x = A -> ( x C_ y <-> A C_ y ) )
20	19	anbi2d 710	. . . . . 6 |- ( x = A -> ( ( y C_ A /\ x C_ y ) <-> ( y C_ A /\ A C_ y ) ) )
21		eqss 3447	. . . . . . 7 |- ( y = A <-> ( y C_ A /\ A C_ y ) )
22	21	biimpri 210	. . . . . 6 |- ( ( y C_ A /\ A C_ y ) -> y = A )
23	20, 22	syl6bi 232	. . . . 5 |- ( x = A -> ( ( y C_ A /\ x C_ y ) -> y = A ) )
24	23	com12 32	. . . 4 |- ( ( y C_ A /\ x C_ y ) -> ( x = A -> y = A ) )
25	24	3adant1 1026	. . 3 |- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ y ) -> ( x = A -> y = A ) )
26		sbcid 3284	. . 3 |- ( [. x / x ]. x = A <-> x = A )
27		vex 3048	. . . 4 |- y e. _V
28		eqsbc3 3307	. . . 4 |- ( y e. _V -> ( [. y / x ]. x = A <-> y = A ) )
29	27, 28	ax-mp 5	. . 3 |- ( [. y / x ]. x = A <-> y = A )
30	25, 26, 29	3imtr4g 274	. 2 |- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ y ) -> ( [. x / x ]. x = A -> [. y / x ]. x = A ) )
31		ineq12 3629	. . . . . 6 |- ( ( y = A /\ x = A ) -> ( y i^i x ) = ( A i^i A ) )
32		inidm 3641	. . . . . 6 |- ( A i^i A ) = A
33	31, 32	syl6eq 2501	. . . . 5 |- ( ( y = A /\ x = A ) -> ( y i^i x ) = A )
34	29, 26, 33	syl2anb 482	. . . 4 |- ( ( [. y / x ]. x = A /\ [. x / x ]. x = A ) -> ( y i^i x ) = A )
35	27	inex1 4544	. . . . 5 |- ( y i^i x ) e. _V
36		eqsbc3 3307	. . . . 5 |- ( ( y i^i x ) e. _V -> ( [. ( y i^i x ) / x ]. x = A <-> ( y i^i x ) = A ) )
37	35, 36	ax-mp 5	. . . 4 |- ( [. ( y i^i x ) / x ]. x = A <-> ( y i^i x ) = A )
38	34, 37	sylibr 216	. . 3 |- ( ( [. y / x ]. x = A /\ [. x / x ]. x = A ) -> [. ( y i^i x ) / x ]. x = A )
39	38	a1i 11	. 2 |- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ A ) -> ( ( [. y / x ]. x = A /\ [. x / x ]. x = A ) -> [. ( y i^i x ) / x ]. x = A ) )
40	5, 7, 11, 18, 30, 39	isfild 20873	1 |- ( ( A e. B /\ A =/= (/) ) -> { A } e. ( Fil ` A ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   _Vcvv 3045   [.wsbc 3267   i^i cin 3403   C_ wss 3404   (/)c0 3731   {csn 3968   ` cfv 5582   Filcfil 20860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-fbas 18967  df-fil 20861

This theorem is referenced by:  snfbas  20881