Theorem discsntermlem 50622

Description: A singlegon is an element of the class of singlegons. The converse (basrestermcfolem 50623) also holds. This is trivial if 𝐵 is 𝑏 (abid 2743). (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 5393	. . . . 5 ⊢ {𝑥} ∈ V
2		eleq1 2849	. . . . 5 ⊢ (𝐵 = {𝑥} → (𝐵 ∈ V ↔ {𝑥} ∈ V))
3	1, 2	mpbiri 261	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 ∈ V)
4	3	exlimiv 1963	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ V)
5		eqeq1 2765	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 = {𝑥} ↔ 𝐵 = {𝑥}))
6	5	exbidv 1954	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑥 𝑏 = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
7	6	elabg 3630	. . 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}})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  Vcvv 3451  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-sn 4585  df-pr 4587

This theorem is used by:  discsnterm  50626  basrestermcfo  50627