Theorem sssn 3142

Description: The subsets of a singleton.

Assertion

Ref	Expression
sssn	|- (A C_ {B} <-> (A = (/) \/ A = {B}))

Proof of Theorem sssn

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . . . . . . . 11 |- (A C_ {B} -> (x e. A -> x e. {B}))
2		elsni 3066	. . . . . . . . . . 11 |- (x e. {B} -> x = B)
3	1, 2	syl6 25	. . . . . . . . . 10 |- (A C_ {B} -> (x e. A -> x = B))
4		eleq1 1957	. . . . . . . . . 10 |- (x = B -> (x e. A <-> B e. A))
5	3, 4	syl6 25	. . . . . . . . 9 |- (A C_ {B} -> (x e. A -> (x e. A <-> B e. A)))
6	5	ibd 654	. . . . . . . 8 |- (A C_ {B} -> (x e. A -> B e. A))
7	6	19.23adv 1584	. . . . . . 7 |- (A C_ {B} -> (E.x x e. A -> B e. A))
8		neq0 2885	. . . . . . 7 |- (-. A = (/) <-> E.x x e. A)
9	7, 8	syl5ib 223	. . . . . 6 |- (A C_ {B} -> (-. A = (/) -> B e. A))
10		snssi 3129	. . . . . 6 |- (B e. A -> {B} C_ A)
11	9, 10	syl6 25	. . . . 5 |- (A C_ {B} -> (-. A = (/) -> {B} C_ A))
12	11	anc2li 326	. . . 4 |- (A C_ {B} -> (-. A = (/) -> (A C_ {B} /\ {B} C_ A)))
13		eqss 2631	. . . 4 |- (A = {B} <-> (A C_ {B} /\ {B} C_ A))
14	12, 13	syl6ibr 230	. . 3 |- (A C_ {B} -> (-. A = (/) -> A = {B}))
15	14	orrd 250	. 2 |- (A C_ {B} -> (A = (/) \/ A = {B}))
16		0ss 2900	. . . 4 |- (/) C_ {B}
17		sseq1 2637	. . . 4 |- (A = (/) -> (A C_ {B} <-> (/) C_ {B}))
18	16, 17	mpbiri 211	. . 3 |- (A = (/) -> A C_ {B})
19		eqimss 2665	. . 3 |- (A = {B} -> A C_ {B})
20	18, 19	jaoi 368	. 2 |- ((A = (/) \/ A = {B}) -> A C_ {B})
21	15, 20	impbii 174	1 |- (A C_ {B} <-> (A = (/) \/ A = {B}))

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   C_ wss 2593  (/)c0 2875  {csn 3044

This theorem is referenced by:  eqsn 3143  sspr 3144  snsssn 3148  pwsn 3172  foconst 4629  0top 8905  sn0top 8917  bnj140 12472  top2usne 14898

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-sn 3049

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