Theorem sneqr 4768

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 4767	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-sn 4559

This theorem is referenced by:  snsssn  4769  mosneq  4770  opth1  5384  propeqop  5415  opthwiener  5422  funsndifnop  7005  canth2  8866  axcc2lem  10123  hashge3el3dif  14128  dis2ndc  22519  axlowdim1  27230  bj-snsetex  35080  poimirlem13  35717  poimirlem14  35718  wopprc  40768  snen1g  41029  mnuprdlem2  41780  hoidmv1le  44022  fsetsnf1  44433