Theorem snss 3828

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3827). 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 3827	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210  {csn 3688

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-sn 3694

This theorem is referenced by:  snssgOLD  3829  prss  3849  tpss  3861  snelpw  4327  sspwb  4331  mss  4341  exss  4342  reg2exmidlema  4655  elomssom  4726  relsn  4854  fnressn  5869  un0mulcl  9529  nn0ssz  9594  hashfibclem  11202  fimaxre2  11908  fsum2dlemstep  12116  fsumabs  12147  fsumiun  12159  fprod2dlemstep  12304  dvmptfsum  15582  elply2  15592  elplyd  15598  ply1term  15600  plymullem  15607  bdsnss  16635