Theorem sneqr 4795

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 4794	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-sn 4580

This theorem is referenced by:  snsssn  4796  mosneq  4797  opth1  5440  propeqop  5473  opthwiener  5480  funsndifnop  7129  canth2  9096  axcc2lem  10387  hashge3el3dif  14494  dis2ndc  23508  axlowdim1  29117  selvply1rhmlema  33776  selvply1rhmlem1  33778  esplyfval1  33831  mh-inf3sn  36863  bj-snsetex  37409  poimirlem13  38093  poimirlem14  38094  wopprc  43568  snen1g  44061  mnuprdlem2  44810  hoidmv1le  47129  fsetsnf1  47607