Theorem basrestermcfolem 49309

Description: An element of the class of singlegons is a singlegon. The converse (discsntermlem 49308) also holds. This is trivial if 𝐵 is 𝑏 (abid 2716). (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 2738	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 = {𝑥} ↔ 𝐵 = {𝑥}))
2	1	exbidv 1920	. . 3 ⊢ (𝑏 = 𝐵 → (∃𝑥 𝑏 = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
3	2	elabg 3653	. 2 ⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} ↔ ∃𝑥 𝐵 = {𝑥}))
4	3	ibi 267	1 ⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → ∃𝑥 𝐵 = {𝑥})

Syntax hints:   → wi 4   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2712  {csn 4599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808

This theorem is referenced by:  basrestermcfo  49313