Theorem basrestermcfolem 50152

Description: An element of the class of singlegons is a singlegon. The converse (discsntermlem 50151) also holds. This is trivial if 𝐵 is 𝑏 (abid 2743). (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 2765	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 = {𝑥} ↔ 𝐵 = {𝑥}))
2	1	exbidv 1940	. . 3 ⊢ (𝑏 = 𝐵 → (∃𝑥 𝑏 = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
3	2	elabg 3634	. 2 ⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} ↔ ∃𝑥 𝐵 = {𝑥}))
4	3	ibi 269	1 ⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → ∃𝑥 𝐵 = {𝑥})

Syntax hints:   → wi 4   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836

This theorem is referenced by:  basrestermcfo  50156