Theorem discsntermlem 49601

Description: A singlegon is an element of the class of singlegons. The converse (basrestermcfolem 49602) also holds. This is trivial if 𝐵 is 𝑏 (abid 2713). (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 5372	. . . . 5 ⊢ {𝑥} ∈ V
2		eleq1 2819	. . . . 5 ⊢ (𝐵 = {𝑥} → (𝐵 ∈ V ↔ {𝑥} ∈ V))
3	1, 2	mpbiri 258	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 ∈ V)
4	3	exlimiv 1931	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ V)
5		eqeq1 2735	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 = {𝑥} ↔ 𝐵 = {𝑥}))
6	5	exbidv 1922	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑥 𝑏 = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
7	6	elabg 3632	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} ↔ ∃𝑥 𝐵 = {𝑥}))
8	4, 7	syl 17	. 2 ⊢ (∃𝑥 𝐵 = {𝑥} → (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} ↔ ∃𝑥 𝐵 = {𝑥}))
9	8	ibir 268	1 ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  Vcvv 3436  {csn 4576

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3907  df-sn 4577  df-pr 4579

This theorem is referenced by:  discsnterm  49605  basrestermcfo  49606