Theorem sneqr 2542

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 2496	. . 3 |- A e. {A}
3		eleq2 1578	. . 3 |- ({A} = {B} -> (A e. {A} <-> A e. {B}))
4	2, 3	mpbii 191	. 2 |- ({A} = {B} -> A e. {B})
5	1	elsnc 2492	. 2 |- (A e. {B} <-> A = B)
6	4, 5	sylib 196	1 |- ({A} = {B} -> A = B)

Syntax hints:   -> wi 3   = wceq 992   e. wcel 994  Vcvv 1857  {csn 2467

This theorem is referenced by:  snsssn 2543  opth2 2876  opthwiener 2884  canth2 4629  ismrer1 12080

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-un 2102  df-sn 2470  df-pr 2471

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