Theorem snss 3834

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3833). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)

Hypothesis

Ref	Expression
snss.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snss	⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Proof of Theorem snss

Step	Hyp	Ref	Expression
1		snss.1	. 2 ⊢ 𝐴 ∈ V
2		snssg 3833	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214  {csn 3694

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-in 3220  df-ss 3227  df-sn 3700

This theorem is referenced by:  snssgOLD  3835  prss  3855  tpss  3867  snelpw  4333  sspwb  4337  mss  4347  exss  4348  reg2exmidlema  4661  elomssom  4732  relsn  4860  fnressn  5875  un0mulcl  9550  nn0ssz  9615  hashfibclem  11234  fimaxre2  11941  fsum2dlemstep  12149  fsumabs  12180  fsumiun  12192  fprod2dlemstep  12337  dvmptfsum  15720  elply2  15730  elplyd  15736  ply1term  15738  plymullem  15745  bdsnss  16783