Theorem elsnd 4609

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 4608	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-sn 4592

This theorem is referenced by:  sndisj  5102  xpdifcnvepel  6164  chnccats1  18677  chnccat  18678  ex-chn1  18689  lnincplng  29020  elrgspnsubrunlem2  33505  0mplrim  33845  selvply1rhmlemb  33850  evlextv  33873  discsubc  49722