Theorem sneqr 4839

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 4838	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3462  {csn 4623

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 4624

This theorem is referenced by:  snsssn  4840  mosneq  4841  opth1  5473  propeqop  5505  opthwiener  5512  funsndifnop  7157  canth2  9160  axcc2lem  10470  hashge3el3dif  14500  dis2ndc  23452  axlowdim1  28890  bj-snsetex  36683  poimirlem13  37347  poimirlem14  37348  wopprc  42725  snen1g  43228  mnuprdlem2  43984  hoidmv1le  46251  fsetsnf1  46703