Theorem sneq 3533

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 2147	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))
2	1	abbidv 2255	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝐵})
3		df-sn 3528	. 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
4		df-sn 3528	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
5	2, 3, 4	3eqtr4g 2195	1 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})

Syntax hints:   → wi 4   = wceq 1331  {cab 2123  {csn 3522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-sn 3528

This theorem is referenced by:  sneqi  3534  sneqd  3535  euabsn  3588  absneu  3590  preq1  3595  tpeq3  3606  snssg  3651  sneqrg  3684  sneqbg  3685  opeq1  3700  unisng  3748  exmidsssn  4120  exmidsssnc  4121  suceq  4319  snnex  4364  opeliunxp  4589  relop  4684  elimasng  4902  dmsnsnsng  5011  elxp4  5021  elxp5  5022  iotajust  5082  fconstg  5314  f1osng  5401  nfvres  5447  fsng  5586  fnressn  5599  fressnfv  5600  funfvima3  5644  isoselem  5714  1stvalg  6033  2ndvalg  6034  2ndval2  6047  fo1st  6048  fo2nd  6049  f1stres  6050  f2ndres  6051  mpomptsx  6088  dmmpossx  6090  fmpox  6091  brtpos2  6141  dftpos4  6153  tpostpos  6154  eceq1  6457  fvdiagfn  6580  mapsncnv  6582  elixpsn  6622  ixpsnf1o  6623  ensn1g  6684  en1  6686  xpsneng  6709  xpcomco  6713  xpassen  6717  xpdom2  6718  phplem3  6741  phplem3g  6743  fidifsnen  6757  xpfi  6811  pm54.43  7039  exp3val  10288  fsum2dlemstep  11196  fsumcnv  11199  fisumcom2  11200  txswaphmeolem  12478