Theorem discsntermlem 50067

Description: A singlegon is an element of the class of singlegons. The converse (basrestermcfolem 50068) also holds. This is trivial if 𝐵 is 𝑏 (abid 2722). (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 5371	. . . . 5 ⊢ {𝑥} ∈ V
2		eleq1 2828	. . . . 5 ⊢ (𝐵 = {𝑥} → (𝐵 ∈ V ↔ {𝑥} ∈ V))
3	1, 2	mpbiri 259	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 ∈ V)
4	3	exlimiv 1937	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ V)
5		eqeq1 2744	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 = {𝑥} ↔ 𝐵 = {𝑥}))
6	5	exbidv 1928	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑥 𝑏 = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
7	6	elabg 3621	. . 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 269	1 ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}})

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2718  Vcvv 3432  {csn 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-sn 4563  df-pr 4565

This theorem is referenced by:  discsnterm  50071  basrestermcfo  50072