Theorem sneqr 4794

Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)

Hypothesis

Ref	Expression
sneqr.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
sneqr	⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Proof of Theorem sneqr

Step	Hyp	Ref	Expression
1		sneqr.1	. 2 ⊢ 𝐴 ∈ V
2		sneqrg 4793	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-sn 4585

This theorem is used by:  snsssn  4795  mosneq  4796  opth1  5443  propeqop  5478  opthwiener  5486  funsndifnop  7158  canth2  9154  axcc2lem  10519  hashge3el3dif  14637  dis2ndc  23786  axlowdim1  29544  selvply1rhmlema  34198  selvply1rhmlem1  34200  esplyfval1  34253  mh-inf3sn  37424  bj-snsetex  37970  poimirlem13  38658  poimirlem14  38659  wopprc  44144  snen1g  44632  mnuprdlem2  45364  hoidmv1le  47711  fsetsnf1  48229