Theorem elsnd 4598

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 4597	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-sn 4585

This theorem is used by:  sndisj  5094  xpdifcnvepel  6161  chnccats1  18806  chnccat  18807  ex-chn1  18818  lnincplng  29269  elrgspnsubrunlem2  33857  0mplrim  34194  selvply1rhmlemb  34199  evlextv  34222  chnerlem1  47999  tmachlem-agreeprod  48054  discsubc  50285  parhomideq2  50856  parhomideq  50857