 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sneqr GIF version

Theorem sneqr 3872
 Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)
Hypothesis
Ref Expression
sneqr.1 A V
Assertion
Ref Expression
sneqr ({A} = {B} → A = B)

Proof of Theorem sneqr
StepHypRef Expression
1 sneqr.1 . . . 4 A V
21snid 3760 . . 3 A {A}
3 eleq2 2414 . . 3 ({A} = {B} → (A {A} ↔ A {B}))
42, 3mpbii 202 . 2 ({A} = {B} → A {B})
51elsnc 3756 . 2 (A {B} ↔ A = B)
64, 5sylib 188 1 ({A} = {B} → A = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sn 3741 This theorem is referenced by:  snsssn  3873  sneqrg  3874
 Copyright terms: Public domain W3C validator