Theorem eqsn 2465

Description: Two ways to express that a nonempty set equals a singleton.

Assertion

Ref	Expression
eqsn	|- (A =/= (/) -> (A = {B} <-> A.x e. A x = B))

Distinct variable groups:   x,A   x,B

Proof of Theorem eqsn

Step	Hyp	Ref	Expression
1		eqimss 2099	. . 3 |- (A = {B} -> A (_ {B})
2		sssn 2464	. . . . . . 7 |- (A (_ {B} <-> (A = (/) \/ A = {B}))
3	2	biimp 151	. . . . . 6 |- (A (_ {B} -> (A = (/) \/ A = {B}))
4	3	ord 232	. . . . 5 |- (A (_ {B} -> (-. A = (/) -> A = {B}))
5		df-ne 1579	. . . . 5 |- (A =/= (/) <-> -. A = (/))
6	4, 5	syl5ib 206	. . . 4 |- (A (_ {B} -> (A =/= (/) -> A = {B}))
7	6	com12 11	. . 3 |- (A =/= (/) -> (A (_ {B} -> A = {B}))
8	1, 7	impbid2 516	. 2 |- (A =/= (/) -> (A = {B} <-> A (_ {B}))
9		dfss3 2049	. . 3 |- (A (_ {B} <-> A.x e. A x e. {B})
10		elsn 2411	. . . 4 |- (x e. {B} <-> x = B)
11	10	ralbii 1659	. . 3 |- (A.x e. A x e. {B} <-> A.x e. A x = B)
12	9, 11	bitr 173	. 2 |- (A (_ {B} <-> A.x e. A x = B)
13	8, 12	syl6bb 534	1 |- (A =/= (/) -> (A = {B} <-> A.x e. A x = B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637   (_ wss 2037  (/)c0 2270  {csn 2399

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-sn 2402  df-pr 2403

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