Theorem sneqr 3147

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 3069	. . 3 |- A e. {A}
3		eleq2 1958	. . 3 |- ({A} = {B} -> (A e. {A} <-> A e. {B}))
4	2, 3	mpbii 210	. 2 |- ({A} = {B} -> A e. {B})
5	1	elsnc 3065	. 2 |- (A e. {B} <-> A = B)
6	4, 5	sylib 215	1 |- ({A} = {B} -> A = B)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044

This theorem is referenced by:  snsssn 3148  opth2 3546  opthwiener 3554  canth2 5548  sneqrg 13822  ismrer1 16024

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-v 2294  df-sn 3049

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