Theorem sneqbg 4807

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 4803	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2		sneq 4598	. 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 1569   ∈ wcel 2142  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-sn 4589

This theorem is used by:  iotaval2  6507  suppval1  8160  suppsnop  8172  fseqdom  10017  infpwfidom  10019  canthwe  10642  s111  14660  initoid  18064  termoid  18065  embedsetcestrclem  18219  mat1dimelbas  22639  mat1dimbas  22640  unidifsnne  32893  selvply1rhmlem2  33920  altopthg  36467  altopthbg  36468  bj-snglc  37633  f1omptsnlem  38010  fvineqsnf1  38084  extid  38993  suceqsneq  39161  qmapeldisjsim  39537  sn-iotalem  43020  eusnsn  47791