Theorem snfil 20534

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 4030	. . . 4 |- ( x e. { A } <-> x = A )
2		eqimss 3541	. . . . 5 |- ( x = A -> x C_ A )
3	2	pm4.71ri 631	. . . 4 |- ( x = A <-> ( x C_ A /\ x = A ) )
4	1, 3	bitri 249	. . 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 3115	. . 3 |- ( A e. B -> A e. _V )
7	6	adantr 463	. 2 |- ( ( A e. B /\ A =/= (/) ) -> A e. _V )
8		eqid 2454	. . . 4 |- A = A
9		eqsbc3 3364	. . . 4 |- ( A e. B -> ( [. A / x ]. x = A <-> A = A ) )
10	8, 9	mpbiri 233	. . 3 |- ( A e. B -> [. A / x ]. x = A )
11	10	adantr 463	. 2 |- ( ( A e. B /\ A =/= (/) ) -> [. A / x ]. x = A )
12		simpr 459	. . . . 5 |- ( ( A e. B /\ A =/= (/) ) -> A =/= (/) )
13	12	necomd 2725	. . . 4 |- ( ( A e. B /\ A =/= (/) ) -> (/) =/= A )
14	13	neneqd 2656	. . 3 |- ( ( A e. B /\ A =/= (/) ) -> -. (/) = A )
15		0ex 4569	. . . 4 |- (/) e. _V
16		eqsbc3 3364	. . . 4 |- ( (/) e. _V -> ( [. (/) / x ]. x = A <-> (/) = A ) )
17	15, 16	ax-mp 5	. . 3 |- ( [. (/) / x ]. x = A <-> (/) = A )
18	14, 17	sylnibr 303	. 2 |- ( ( A e. B /\ A =/= (/) ) -> -. [. (/) / x ]. x = A )
19		sseq1 3510	. . . . . . 7 |- ( x = A -> ( x C_ y <-> A C_ y ) )
20	19	anbi2d 701	. . . . . 6 |- ( x = A -> ( ( y C_ A /\ x C_ y ) <-> ( y C_ A /\ A C_ y ) ) )
21		eqss 3504	. . . . . . 7 |- ( y = A <-> ( y C_ A /\ A C_ y ) )
22	21	biimpri 206	. . . . . 6 |- ( ( y C_ A /\ A C_ y ) -> y = A )
23	20, 22	syl6bi 228	. . . . 5 |- ( x = A -> ( ( y C_ A /\ x C_ y ) -> y = A ) )
24	23	com12 31	. . . 4 |- ( ( y C_ A /\ x C_ y ) -> ( x = A -> y = A ) )
25	24	3adant1 1012	. . 3 |- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ y ) -> ( x = A -> y = A ) )
26		sbcid 3341	. . 3 |- ( [. x / x ]. x = A <-> x = A )
27		vex 3109	. . . 4 |- y e. _V
28		eqsbc3 3364	. . . 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 270	. 2 |- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ y ) -> ( [. x / x ]. x = A -> [. y / x ]. x = A ) )
31		ineq12 3681	. . . . . 6 |- ( ( y = A /\ x = A ) -> ( y i^i x ) = ( A i^i A ) )
32		inidm 3693	. . . . . 6 |- ( A i^i A ) = A
33	31, 32	syl6eq 2511	. . . . 5 |- ( ( y = A /\ x = A ) -> ( y i^i x ) = A )
34	29, 26, 33	syl2anb 477	. . . 4 |- ( ( [. y / x ]. x = A /\ [. x / x ]. x = A ) -> ( y i^i x ) = A )
35	27	inex1 4578	. . . . 5 |- ( y i^i x ) e. _V
36		eqsbc3 3364	. . . . 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 212	. . 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 20528	1 |- ( ( A e. B /\ A =/= (/) ) -> { A } e. ( Fil ` A ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   _Vcvv 3106   [.wsbc 3324   i^i cin 3460   C_ wss 3461   (/)c0 3783   {csn 4016   ` cfv 5570   Filcfil 20515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-fbas 18614  df-fil 20516

This theorem is referenced by:  snfbas  20536