Theorem discsntermlem 49695

Description: A singlegon is an element of the class of singlegons. The converse (basrestermcfolem 49696) also holds. This is trivial if 𝐵 is 𝑏 (abid 2715). (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 5374	. . . . 5 ⊢ {𝑥} ∈ V
2		eleq1 2821	. . . . 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 2737	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 = {𝑥} ↔ 𝐵 = {𝑥}))
6	5	exbidv 1922	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑥 𝑏 = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
7	6	elabg 3628	. . 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 2113  {cab 2711  Vcvv 3437  {csn 4575

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-pr 5372

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 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903  df-sn 4576  df-pr 4578

This theorem is referenced by:  discsnterm  49699  basrestermcfo  49700