Theorem snmb 3755

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 2779	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		velsn 3651	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	exbii 1629	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
4	1, 3	bitr4i 187	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 ∈ {𝐴})

Syntax hints:   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773  {csn 3634

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sn 3640

This theorem is referenced by:  lpvtx  15719