Theorem snmb 3832

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 2828	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		velsn 3725	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	exbii 1658	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
4	1, 3	bitr4i 187	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 ∈ {𝐴})

Syntax hints:   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  {csn 3708

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3714

This theorem is referenced by:  lpvtx  16303