Theorem sneqr 4778

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 4777	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3432  {csn 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-sn 4563

This theorem is referenced by:  snsssn  4779  mosneq  4780  opth1  5422  propeqop  5455  opthwiener  5462  funsndifnop  7101  canth2  9065  axcc2lem  10356  hashge3el3dif  14447  dis2ndc  23450  axlowdim1  29053  selvply1rhmlema  33709  selvply1rhmlem1  33711  esplyfval1  33764  mh-inf3sn  36777  bj-snsetex  37323  poimirlem13  38007  poimirlem14  38008  wopprc  43482  snen1g  43975  mnuprdlem2  44724  hoidmv1le  47044  fsetsnf1  47522