Theorem sneq 3411

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 2091	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))
2	1	abbidv 2197	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝐵})
3		df-sn 3406	. 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
4		df-sn 3406	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
5	2, 3, 4	3eqtr4g 2139	1 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})

Syntax hints:   → wi 4   = wceq 1285  {cab 2068  {csn 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-sn 3406

This theorem is referenced by:  sneqi  3412  sneqd  3413  euabsn  3464  absneu  3466  preq1  3471  tpeq3  3482  snssg  3524  sneqrg  3556  sneqbg  3557  opeq1  3572  unisng  3620  suceq  4159  snnex  4201  opeliunxp  4415  relop  4508  elimasng  4717  dmsnsnsng  4822  elxp4  4832  elxp5  4833  iotajust  4890  fconstg  5108  f1osng  5192  nfvres  5232  fsng  5362  fnressn  5375  fressnfv  5376  funfvima3  5418  isoselem  5484  1stvalg  5794  2ndvalg  5795  2ndval2  5808  fo1st  5809  fo2nd  5810  f1stres  5811  f2ndres  5812  mpt2mptsx  5848  dmmpt2ssx  5850  fmpt2x  5851  brtpos2  5894  dftpos4  5906  tpostpos  5907  eceq1  6200  ensn1g  6336  en1  6338  xpsneng  6356  xpcomco  6360  xpassen  6364  xpdom2  6365  phplem3  6379  phplem3g  6381  fidifsnen  6395  pm54.43  6508  expival  9564