Theorem snmb 3815

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 2822	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		velsn 3708	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	exbii 1654	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
4	1, 3	bitr4i 187	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 ∈ {𝐴})

Syntax hints:   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815  {csn 3691

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-sn 3697

This theorem is referenced by:  lpvtx  16123