Theorem discsntermlem 50325

Description: A singlegon is an element of the class of singlegons. The converse (basrestermcfolem 50326) also holds. This is trivial if 𝐵 is 𝑏 (abid 2745). (Contributed by Zhi Wang, 20-Oct-2025.)

Assertion

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

Distinct variable group:   𝐵,𝑏,𝑥

Proof of Theorem discsntermlem

Step	Hyp	Ref	Expression
1		vsnex 5408	. . . . 5 ⊢ {𝑥} ∈ V
2		eleq1 2851	. . . . 5 ⊢ (𝐵 = {𝑥} → (𝐵 ∈ V ↔ {𝑥} ∈ V))
3	1, 2	mpbiri 261	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 ∈ V)
4	3	exlimiv 1960	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ V)
5		eqeq1 2767	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 = {𝑥} ↔ 𝐵 = {𝑥}))
6	5	exbidv 1951	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑥 𝑏 = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
7	6	elabg 3636	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} ↔ ∃𝑥 𝐵 = {𝑥}))
8	4, 7	syl 18	. 2 ⊢ (∃𝑥 𝐵 = {𝑥} → (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} ↔ ∃𝑥 𝐵 = {𝑥}))
9	8	ibir 271	1 ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}})

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  Vcvv 3455  {csn 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3911  df-sn 4591  df-pr 4593

This theorem is referenced by:  discsnterm  50329  basrestermcfo  50330