Theorem sneqr 4783

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 4782	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429  {csn 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-sn 4568

This theorem is referenced by:  snsssn  4784  mosneq  4785  opth1  5428  propeqop  5461  opthwiener  5468  funsndifnop  7105  canth2  9068  axcc2lem  10358  hashge3el3dif  14449  dis2ndc  23425  axlowdim1  29028  esplyfval1  33717  mh-inf3sn  36724  bj-snsetex  37270  poimirlem13  37954  poimirlem14  37955  wopprc  43458  snen1g  43951  mnuprdlem2  44700  hoidmv1le  47022  fsetsnf1  47500