Theorem sneqr 2468

Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15.

Hypothesis

Ref	Expression
sneqr.1	|- A e. V

Assertion

Ref	Expression
sneqr	|- ({A} = {B} -> A = B)

Proof of Theorem sneqr

Step	Hyp	Ref	Expression
1		sneqr.1	. . . 4 |- A e. V
2	1	snid 2425	. . 3 |- A e. {A}
3		eleq2 1527	. . 3 |- ({A} = {B} -> (A e. {A} <-> A e. {B}))
4	2, 3	mpbii 193	. 2 |- ({A} = {B} -> A e. {B})
5	1	elsnc 2421	. 2 |- (A e. {B} <-> A = B)
6	4, 5	sylib 198	1 |- ({A} = {B} -> A = B)

Syntax hints:   -> wi 3   = wceq 953   e. wcel 955  Vcvv 1802  {csn 2399

This theorem is referenced by:  snsssn 2469  opth2 2789  opthwiener 2796  canth2 4464

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-v 1803  df-un 2040  df-sn 2402  df-pr 2403

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