Theorem sneq 3684

Description: Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sneq	|- ( A = B -> { A } = { B } )

Proof of Theorem sneq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2241	. . 3 |- ( A = B -> ( x = A <-> x = B ) )
2	1	abbidv 2350	. 2 |- ( A = B -> { x | x = A } = { x | x = B } )
3		df-sn 3679	. 2 |- { A } = { x | x = A }
4		df-sn 3679	. 2 |- { B } = { x | x = B }
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> { A } = { B } )

Syntax hints:   -> wi 4   = wceq 1398   {cab 2217   {csn 3673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-sn 3679

This theorem is referenced by:  sneqi  3685  sneqd  3686  euabsn  3745  absneu  3747  preq1  3752  tpeq3  3763  snssgOLD  3814  sneqrg  3850  sneqbg  3851  opeq1  3867  unisng  3915  exmidsssn  4298  exmidsssnc  4299  suceq  4505  snnex  4551  opeliunxp  4787  relop  4886  elimasng  5111  dmsnsnsng  5221  elxp4  5231  elxp5  5232  iotajust  5292  fconstg  5542  f1osng  5635  nfvres  5684  fsng  5828  fsn2g  5830  funopsn  5838  fnressn  5848  fressnfv  5849  funfvima3  5898  isoselem  5971  1stvalg  6314  2ndvalg  6315  2ndval2  6328  fo1st  6329  fo2nd  6330  f1stres  6331  f2ndres  6332  mpomptsx  6371  dmmpossx  6373  fmpox  6374  suppval  6415  suppsnopdc  6428  brtpos2  6460  dftpos4  6472  tpostpos  6473  eceq1  6780  fvdiagfn  6905  mapsncnv  6907  elixpsn  6947  ixpsnf1o  6948  ensn1g  7014  en1  7016  xpsneng  7049  xpcomco  7053  xpassen  7057  xpdom2  7058  phplem3  7083  phplem3g  7085  fidifsnen  7100  xpfi  7167  pm54.43  7455  cc2lem  7545  cc2  7546  exp3val  10866  fsum2dlemstep  12075  fsumcnv  12078  fisumcom2  12079  fprod2dlemstep  12263  fprodcnv  12266  fprodcom2fi  12267  pwsval  13454  lssats2  14510  lspsneq0  14522  txswaphmeolem  15131  vtxdgfifival  16232  vtxdumgrfival  16239  1loopgrvd2fi  16246  wlk1walkdom  16300  wlkres  16320  eupth2lem3lem3fi  16411