Theorem snfil 21668

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		velsn 4193	. . . 4 |- ( x e. { A } <-> x = A )
2		eqimss 3657	. . . . 5 |- ( x = A -> x C_ A )
3	2	pm4.71ri 665	. . . 4 |- ( x = A <-> ( x C_ A /\ x = A ) )
4	1, 3	bitri 264	. . 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 3212	. . 3 |- ( A e. B -> A e. _V )
7	6	adantr 481	. 2 |- ( ( A e. B /\ A =/= (/) ) -> A e. _V )
8		eqid 2622	. . . 4 |- A = A
9		eqsbc3 3475	. . . 4 |- ( A e. B -> ( [. A / x ]. x = A <-> A = A ) )
10	8, 9	mpbiri 248	. . 3 |- ( A e. B -> [. A / x ]. x = A )
11	10	adantr 481	. 2 |- ( ( A e. B /\ A =/= (/) ) -> [. A / x ]. x = A )
12		simpr 477	. . . . 5 |- ( ( A e. B /\ A =/= (/) ) -> A =/= (/) )
13	12	necomd 2849	. . . 4 |- ( ( A e. B /\ A =/= (/) ) -> (/) =/= A )
14	13	neneqd 2799	. . 3 |- ( ( A e. B /\ A =/= (/) ) -> -. (/) = A )
15		0ex 4790	. . . 4 |- (/) e. _V
16		eqsbc3 3475	. . . 4 |- ( (/) e. _V -> ( [. (/) / x ]. x = A <-> (/) = A ) )
17	15, 16	ax-mp 5	. . 3 |- ( [. (/) / x ]. x = A <-> (/) = A )
18	14, 17	sylnibr 319	. 2 |- ( ( A e. B /\ A =/= (/) ) -> -. [. (/) / x ]. x = A )
19		sseq1 3626	. . . . . . 7 |- ( x = A -> ( x C_ y <-> A C_ y ) )
20	19	anbi2d 740	. . . . . 6 |- ( x = A -> ( ( y C_ A /\ x C_ y ) <-> ( y C_ A /\ A C_ y ) ) )
21		eqss 3618	. . . . . . 7 |- ( y = A <-> ( y C_ A /\ A C_ y ) )
22	21	biimpri 218	. . . . . 6 |- ( ( y C_ A /\ A C_ y ) -> y = A )
23	20, 22	syl6bi 243	. . . . 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 1079	. . 3 |- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ y ) -> ( x = A -> y = A ) )
26		sbcid 3452	. . 3 |- ( [. x / x ]. x = A <-> x = A )
27		vex 3203	. . . 4 |- y e. _V
28		eqsbc3 3475	. . . 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 285	. 2 |- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ y ) -> ( [. x / x ]. x = A -> [. y / x ]. x = A ) )
31		ineq12 3809	. . . . . 6 |- ( ( y = A /\ x = A ) -> ( y i^i x ) = ( A i^i A ) )
32		inidm 3822	. . . . . 6 |- ( A i^i A ) = A
33	31, 32	syl6eq 2672	. . . . 5 |- ( ( y = A /\ x = A ) -> ( y i^i x ) = A )
34	29, 26, 33	syl2anb 496	. . . 4 |- ( ( [. y / x ]. x = A /\ [. x / x ]. x = A ) -> ( y i^i x ) = A )
35	27	inex1 4799	. . . . 5 |- ( y i^i x ) e. _V
36		eqsbc3 3475	. . . . 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 224	. . 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 21662	1 |- ( ( A e. B /\ A =/= (/) ) -> { A } e. ( Fil ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   ` cfv 5888   Filcfil 21649

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-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-fbas 19743  df-fil 21650

This theorem is referenced by:  snfbas  21670