Theorem sneqr 4800

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 4799	. 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 2145  Vcvv 3451  {csn 4584

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-sn 4585

This theorem is used by:  snsssn  4801  mosneq  4802  opth1  5444  propeqop  5479  opthwiener  5487  funsndifnop  7153  canth2  9142  axcc2lem  10507  hashge3el3dif  14625  dis2ndc  23772  axlowdim1  29530  selvply1rhmlema  34143  selvply1rhmlem1  34145  esplyfval1  34198  mh-inf3sn  37310  bj-snsetex  37856  poimirlem13  38531  poimirlem14  38532  wopprc  44016  snen1g  44509  mnuprdlem2  45242  hoidmv1le  47573  fsetsnf1  48091