Theorem sneqr 2481

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 2439	. . 3 |- A e. {A}
3		eleq2 1538	. . 3 |- ({A} = {B} -> (A e. {A} <-> A e. {B}))
4	2, 3	mpbii 193	. 2 |- ({A} = {B} -> A e. {B})
5	1	elsnc 2435	. 2 |- (A e. {B} <-> A = B)
6	4, 5	sylib 198	1 |- ({A} = {B} -> A = B)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  Vcvv 1814  {csn 2413

This theorem is referenced by:  snsssn 2482  opth2 2806  opthwiener 2813  canth2 4490

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-v 1815  df-un 2053  df-sn 2416  df-pr 2417

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