Theorem eqsn 2478

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 2112	. . 3 |- (A = {B} -> A (_ {B})
2		sssn 2477	. . . . . . 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 1590	. . . . 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 520	. 2 |- (A =/= (/) -> (A = {B} <-> A (_ {B}))
9		dfss3 2062	. . 3 |- (A (_ {B} <-> A.x e. A x e. {B})
10		elsn 2425	. . . 4 |- (x e. {B} <-> x = B)
11	10	ralbii 1670	. . 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 538	1 |- (A =/= (/) -> (A = {B} <-> A.x e. A x = B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648   (_ wss 2050  (/)c0 2283  {csn 2413

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417

Copyright terms: Public domain