Theorem elsnd 4607

Description: There is at most one element in a singleton. (Contributed by Thierry Arnoux, 13-Oct-2025.)

Hypothesis

Ref	Expression
elsnd.1	⊢ (𝜑 → 𝐴 ∈ {𝐵})

Assertion

Ref	Expression
elsnd	⊢ (𝜑 → 𝐴 = 𝐵)

Proof of Theorem elsnd

Step	Hyp	Ref	Expression
1		elsnd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝐵})
2		elsni 4606	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-sn 4590

This theorem is referenced by:  sndisj  5101  xpdifcnvepel  6166  chnccats1  18676  chnccat  18677  ex-chn1  18688  lnincplng  29066  elrgspnsubrunlem2  33568  0mplrim  33904  selvply1rhmlemb  33909  evlextv  33932  discsubc  49842