Theorem sssn 1852

Description: The only subsets of a singleton are the singleton and the empty set.

Assertion

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

Proof of Theorem sssn

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . . . . 11 |- (A (_ {B} -> (x e. A -> x e. {B}))
2		elsni 1827	. . . . . . . . . . 11 |- (x e. {B} -> x = B)
3	1, 2	syl6 23	. . . . . . . . . 10 |- (A (_ {B} -> (x e. A -> x = B))
4		eleq1 1149	. . . . . . . . . 10 |- (x = B -> (x e. A <-> B e. A))
5	3, 4	syl6 23	. . . . . . . . 9 |- (A (_ {B} -> (x e. A -> (x e. A <-> B e. A)))
6	5	ibd 451	. . . . . . . 8 |- (A (_ {B} -> (x e. A -> B e. A))
7	6	19.23adv 954	. . . . . . 7 |- (A (_ {B} -> (E.x x e. A -> B e. A))
8		n0 1714	. . . . . . 7 |- (-. A = (/) <-> E.x x e. A)
9	7, 8	syl5ib 181	. . . . . 6 |- (A (_ {B} -> (-. A = (/) -> B e. A))
10		snssi 1851	. . . . . 6 |- (B e. A -> {B} (_ A)
11	9, 10	syl6 23	. . . . 5 |- (A (_ {B} -> (-. A = (/) -> {B} (_ A))
12	11	anc2li 250	. . . 4 |- (A (_ {B} -> (-. A = (/) -> (A (_ {B} /\ {B} (_ A)))
13		eqss 1516	. . . 4 |- (A = {B} <-> (A (_ {B} /\ {B} (_ A))
14	12, 13	syl6ibr 186	. . 3 |- (A (_ {B} -> (-. A = (/) -> A = {B}))
15	14	orrd 203	. 2 |- (A (_ {B} -> (A = (/) \/ A = {B}))
16		0ss 1725	. . . 4 |- (/) (_ {B}
17		sseq1 1521	. . . 4 |- (A = (/) -> (A (_ {B} <-> (/) (_ {B}))
18	16, 17	mpbiri 169	. . 3 |- (A = (/) -> A (_ {B})
19		eqimss 1548	. . 3 |- (A = {B} -> A (_ {B})
20	18, 19	jaoi 275	. 2 |- ((A = (/) \/ A = {B}) -> A (_ {B})
21	15, 20	impbi 139	1 |- (A (_ {B} <-> (A = (/) \/ A = {B}))

Syntax hints:  -. wn 1   <-> wb 127   \/ wo 195   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092   (_ wss 1487  (/)c0 1707  {csn 1808

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812

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