Theorem elsnd 4602

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 4601	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sn 4589

This theorem is used by:  sndisj  5100  xpdifcnvepel  6178  chnccats1  18849  chnccat  18850  ex-chn1  18861  lnincplng  29319  elrgspnsubrunlem2  33908  0mplrim  34257  selvply1rhmlemb  34262  evlextv  34285  chnerlem1  48165  tmachlem-agreeprod  48220  discsubc  50509  parhomideq2  51080  parhomideq  51081