Theorem sneqr 4816

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 4815	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459  {csn 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-sn 4602

This theorem is referenced by:  snsssn  4817  mosneq  4818  opth1  5450  propeqop  5482  opthwiener  5489  funsndifnop  7141  canth2  9144  axcc2lem  10450  hashge3el3dif  14505  dis2ndc  23398  axlowdim1  28938  bj-snsetex  36981  poimirlem13  37657  poimirlem14  37658  wopprc  43054  snen1g  43548  mnuprdlem2  44297  hoidmv1le  46623  fsetsnf1  47081