Theorem snmb 3793

Description: A singleton is inhabited iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) (Revised by Jim Kingdon, 29-Dec-2025.)

Assertion

Ref	Expression
snmb	⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 ∈ {𝐴})

Distinct variable group:   𝑥,𝐴

Proof of Theorem snmb

Step	Hyp	Ref	Expression
1		isset 2809	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		velsn 3686	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	exbii 1653	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
4	1, 3	bitr4i 187	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 ∈ {𝐴})

Syntax hints:   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802  {csn 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sn 3675

This theorem is referenced by:  lpvtx  15933