Theorem sneqr 2473

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 2431	. . 3 |- A e. {A}
3		eleq2 1532	. . 3 |- ({A} = {B} -> (A e. {A} <-> A e. {B}))
4	2, 3	mpbii 193	. 2 |- ({A} = {B} -> A e. {B})
5	1	elsnc 2427	. 2 |- (A e. {B} <-> A = B)
6	4, 5	sylib 198	1 |- ({A} = {B} -> A = B)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  Vcvv 1807  {csn 2405

This theorem is referenced by:  snsssn 2474  opth2 2795  opthwiener 2802  canth2 4470

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409

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