Theorem elsnd 4595

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 4594	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {csn 4577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-sn 4578

This theorem is referenced by:  chnccats1  18539  chnccat  18540  ex-chn1  18551  elrgspnsubrunlem2  33258  evlextv  33635  discsubc  49225