 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sneqbg Structured version   GIF version

Theorem sneqbg 3525
 Description: Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
sneqbg (A 𝑉 → ({A} = {B} ↔ A = B))

Proof of Theorem sneqbg
StepHypRef Expression
1 sneqrg 3524 . 2 (A 𝑉 → ({A} = {B} → A = B))
2 sneq 3378 . 2 (A = B → {A} = {B})
31, 2impbid1 130 1 (A 𝑉 → ({A} = {B} ↔ A = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sn 3373 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator