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Theorem sneqr 4555
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
StepHypRef Expression
1 sneqr.1 . 2 𝐴 ∈ V
2 sneqrg 4554 . 2 (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
31, 2ax-mp 5 1 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  Vcvv 3383  {csn 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-sn 4367
This theorem is referenced by:  snsssn  4556  mosneq  4557  opth1  5132  propeqop  5161  opthwiener  5168  funsndifnop  6642  canth2  8353  axcc2lem  9544  hashge3el3dif  13513  dis2ndc  21589  axlowdim1  26188  bj-snsetex  33435  poimirlem13  33903  poimirlem14  33904  wopprc  38370  hoidmv1le  41542
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