Theorem sneqr 4807

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 4806	. 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 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-sn 4592

This theorem is used by:  snsssn  4808  mosneq  4809  opth1  5459  propeqop  5492  opthwiener  5499  funsndifnop  7154  canth2  9125  axcc2lem  10435  hashge3el3dif  14542  dis2ndc  23668  axlowdim1  29364  selvply1rhmlema  33972  selvply1rhmlem1  33974  esplyfval1  34027  mh-inf3sn  37110  bj-snsetex  37656  poimirlem13  38341  poimirlem14  38342  wopprc  43815  snen1g  44308  mnuprdlem2  45041  hoidmv1le  47366  fsetsnf1  47847