Theorem sneq 3654

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 2217	. . 3 |- ( A = B -> ( x = A <-> x = B ) )
2	1	abbidv 2325	. 2 |- ( A = B -> { x | x = A } = { x | x = B } )
3		df-sn 3649	. 2 |- { A } = { x | x = A }
4		df-sn 3649	. 2 |- { B } = { x | x = B }
5	2, 3, 4	3eqtr4g 2265	1 |- ( A = B -> { A } = { B } )

Syntax hints:   -> wi 4   = wceq 1373   {cab 2193   {csn 3643

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-sn 3649

This theorem is referenced by:  sneqi  3655  sneqd  3656  euabsn  3713  absneu  3715  preq1  3720  tpeq3  3731  snssgOLD  3780  sneqrg  3816  sneqbg  3817  opeq1  3833  unisng  3881  exmidsssn  4262  exmidsssnc  4263  suceq  4467  snnex  4513  opeliunxp  4748  relop  4846  elimasng  5069  dmsnsnsng  5179  elxp4  5189  elxp5  5190  iotajust  5250  fconstg  5494  f1osng  5586  nfvres  5633  fsng  5776  funopsn  5785  fnressn  5793  fressnfv  5794  funfvima3  5841  isoselem  5912  1stvalg  6251  2ndvalg  6252  2ndval2  6265  fo1st  6266  fo2nd  6267  f1stres  6268  f2ndres  6269  mpomptsx  6306  dmmpossx  6308  fmpox  6309  brtpos2  6360  dftpos4  6372  tpostpos  6373  eceq1  6678  fvdiagfn  6803  mapsncnv  6805  elixpsn  6845  ixpsnf1o  6846  ensn1g  6912  en1  6914  xpsneng  6942  xpcomco  6946  xpassen  6950  xpdom2  6951  phplem3  6976  phplem3g  6978  fidifsnen  6993  xpfi  7055  pm54.43  7324  cc2lem  7413  cc2  7414  exp3val  10723  fsum2dlemstep  11860  fsumcnv  11863  fisumcom2  11864  fprod2dlemstep  12048  fprodcnv  12051  fprodcom2fi  12052  pwsval  13238  lssats2  14291  lspsneq0  14303  txswaphmeolem  14907