Theorem sneqbg 4843

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 4839	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2		sneq 4634	. 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
3	1, 2	impbid1 224	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099  {csn 4624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sn 4625

This theorem is referenced by:  iotaval2  6512  suppval1  8170  suppsnop  8182  fseqdom  10060  infpwfidom  10062  canthwe  10683  s111  14616  initoid  18016  termoid  18017  embedsetcestrclem  18174  mat1dimelbas  22459  mat1dimbas  22460  unidifsnne  32460  altopthg  35802  altopthbg  35803  bj-snglc  36687  f1omptsnlem  37054  fvineqsnf1  37128  suceqsneq  37945  extid  38019  sn-iotalem  41963  eusnsn  46675