Theorem sneqbg 4801

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

Step	Hyp	Ref	Expression
1		sneqrg 4797	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2		sneq 4594	. 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
3	1, 2	impbid1 228	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sn 4589

This theorem is used by:  iotaval2  6524  suppval1  8208  suppsnop  8220  fseqdom  10129  infpwfidom  10131  canthwe  10760  s111  14788  initoid  18202  termoid  18203  embedsetcestrclem  18357  mat1dimelbas  22813  mat1dimbas  22814  unidifsnne  33201  selvply1rhmlem2  34220  altopthg  36909  altopthbg  36910  bj-snglc  38059  f1omptsnlem  38450  fvineqsnf1  38524  extid  39439  suceqsneq  39607  qmapeldisjsim  39983  sn-iotalem  43466  eusnsn  48299