Theorem sneqr 4793

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 4792	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  {csn 4577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-sn 4578

This theorem is referenced by:  snsssn  4794  mosneq  4795  opth1  5420  propeqop  5452  opthwiener  5459  funsndifnop  7093  canth2  9054  axcc2lem  10338  hashge3el3dif  14401  dis2ndc  23395  axlowdim1  28958  bj-snsetex  37080  poimirlem13  37746  poimirlem14  37747  wopprc  43187  snen1g  43681  mnuprdlem2  44430  hoidmv1le  46754  fsetsnf1  47214