Theorem oefil2 10567

Description: A singleton is a filter. Bourbaki TG I.36, example 1.

Assertion

Ref	Expression
oefil2	|- ((A e. B /\ A =/= (/)) -> {A} e. Fil)

Proof of Theorem oefil2

Step	Hyp	Ref	Expression
1		elsni 2432	. . . . . . 7 |- ((/) e. {A} -> (/) = A)
2	1	eqcomd 1480	. . . . . 6 |- ((/) e. {A} -> A = (/))
3	2	necon3ai 1606	. . . . 5 |- (A =/= (/) -> -. (/) e. {A})
4	3	adantl 388	. . . 4 |- ((A e. B /\ A =/= (/)) -> -. (/) e. {A})
5		unisng 2518	. . . . . 6 |- (A e. B -> U.{A} = A)
6	5	adantr 389	. . . . 5 |- ((A e. B /\ A =/= (/)) -> U.{A} = A)
7		snidg 2433	. . . . . 6 |- (A e. B -> A e. {A})
8	7	adantr 389	. . . . 5 |- ((A e. B /\ A =/= (/)) -> A e. {A})
9	6, 8	eqeltrd 1548	. . . 4 |- ((A e. B /\ A =/= (/)) -> U.{A} e. {A})
10	4, 9	jca 288	. . 3 |- ((A e. B /\ A =/= (/)) -> (-. (/) e. {A} /\ U.{A} e. {A}))
11	5	sseq2d 2089	. . . . . . . . 9 |- (A e. B -> (y (_ U.{A} <-> y (_ A))
12		sseq1 2082	. . . . . . . . . . 11 |- (x = A -> (x (_ y <-> A (_ y))
13		eqss 2077	. . . . . . . . . . . . . . 15 |- (y = A <-> (y (_ A /\ A (_ y))
14	13	biimpr 152	. . . . . . . . . . . . . 14 |- ((y (_ A /\ A (_ y) -> y = A)
15	14	ancoms 436	. . . . . . . . . . . . 13 |- ((A (_ y /\ y (_ A) -> y = A)
16		elsn 2421	. . . . . . . . . . . . 13 |- (y e. {A} <-> y = A)
17	15, 16	sylibr 200	. . . . . . . . . . . 12 |- ((A (_ y /\ y (_ A) -> y e. {A})
18	17	ex 373	. . . . . . . . . . 11 |- (A (_ y -> (y (_ A -> y e. {A}))
19	12, 18	syl6bi 214	. . . . . . . . . 10 |- (x = A -> (x (_ y -> (y (_ A -> y e. {A})))
20	19	com3r 35	. . . . . . . . 9 |- (y (_ A -> (x = A -> (x (_ y -> y e. {A})))
21	11, 20	syl6bi 214	. . . . . . . 8 |- (A e. B -> (y (_ U.{A} -> (x = A -> (x (_ y -> y e. {A}))))
22	21	com23 32	. . . . . . 7 |- (A e. B -> (x = A -> (y (_ U.{A} -> (x (_ y -> y e. {A}))))
23		elsn 2421	. . . . . . 7 |- (x e. {A} <-> x = A)
24	22, 23	syl5ib 206	. . . . . 6 |- (A e. B -> (x e. {A} -> (y (_ U.{A} -> (x (_ y -> y e. {A}))))
25	24	3impd 847	. . . . 5 |- (A e. B -> ((x e. {A} /\ y (_ U.{A} /\ x (_ y) -> y e. {A}))
26	25	19.21aivv 1287	. . . 4 |- (A e. B -> A.xA.y((x e. {A} /\ y (_ U.{A} /\ x (_ y) -> y e. {A}))
27	26	adantr 389	. . 3 |- ((A e. B /\ A =/= (/)) -> A.xA.y((x e. {A} /\ y (_ U.{A} /\ x (_ y) -> y e. {A}))
28		ineq12 2212	. . . . . . . 8 |- ((x = A /\ y = A) -> (x i^i y) = (A i^i A))
29		inidm 2222	. . . . . . . 8 |- (A i^i A) = A
30	28, 29	syl6eq 1523	. . . . . . 7 |- ((x = A /\ y = A) -> (x i^i y) = A)
31		visset 1813	. . . . . . . . 9 |- x e. V
32	31	inex1 2716	. . . . . . . 8 |- (x i^i y) e. V
33	32	elsnc 2431	. . . . . . 7 |- ((x i^i y) e. {A} <-> (x i^i y) = A)
34	30, 33	sylibr 200	. . . . . 6 |- ((x = A /\ y = A) -> (x i^i y) e. {A})
35	34, 23, 16	syl2anb 455	. . . . 5 |- ((x e. {A} /\ y e. {A}) -> (x i^i y) e. {A})
36	35	rgen2a 1699	. . . 4 |- A.x e. {A}A.y e. {A} (x i^i y) e. {A}
37	36	a1i 8	. . 3 |- ((A e. B /\ A =/= (/)) -> A.x e. {A}A.y e. {A} (x i^i y) e. {A})
38	10, 27, 37	3jca 819	. 2 |- ((A e. B /\ A =/= (/)) -> ((-. (/) e. {A} /\ U.{A} e. {A}) /\ A.xA.y((x e. {A} /\ y (_ U.{A} /\ x (_ y) -> y e. {A}) /\ A.x e. {A}A.y e. {A} (x i^i y) e. {A}))
39		snex 2750	. . 3 |- {A} e. V
40		eqid 1475	. . . 4 |- U.{A} = U.{A}
41	40	isfil 10558	. . 3 |- ({A} e. V -> ({A} e. Fil <-> ((-. (/) e. {A} /\ U.{A} e. {A}) /\ A.xA.y((x e. {A} /\ y (_ U.{A} /\ x (_ y) -> y e. {A}) /\ A.x e. {A}A.y e. {A} (x i^i y) e. {A})))
42	39, 41	ax-mp 7	. 2 |- ({A} e. Fil <-> ((-. (/) e. {A} /\ U.{A} e. {A}) /\ A.xA.y((x e. {A} /\ y (_ U.{A} /\ x (_ y) -> y e. {A}) /\ A.x e. {A}A.y e. {A} (x i^i y) e. {A}))
43	38, 42	sylibr 200	1 |- ((A e. B /\ A =/= (/)) -> {A} e. Fil)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  Vcvv 1811   i^i cin 2046   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503  Filcfil 10556

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504  df-fil 10557

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