Theorem basrestermcfolem 50924

Description: An element of the class of singlegons is a singlegon. The converse (discsntermlem 50923) also holds. This is trivial if 𝐵 is 𝑏 (abid 2745). (Contributed by Zhi Wang, 20-Oct-2025.)

Assertion

Ref	Expression
basrestermcfolem	⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → ∃𝑥 𝐵 = {𝑥})

Distinct variable group:   𝐵,𝑏,𝑥

Proof of Theorem basrestermcfolem

Step	Hyp	Ref	Expression
1		eqeq1 2767	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 = {𝑥} ↔ 𝐵 = {𝑥}))
2	1	exbidv 1956	. . 3 ⊢ (𝑏 = 𝐵 → (∃𝑥 𝑏 = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
3	2	elabg 3633	. 2 ⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} ↔ ∃𝑥 𝐵 = {𝑥}))
4	3	ibi 270	1 ⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → ∃𝑥 𝐵 = {𝑥})

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  {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

This theorem is used by:  basrestermcfo  50928