Theorem snss 3848

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3847). 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 3847	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  {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-in 3226  df-ss 3233  df-sn 3714

This theorem is referenced by:  snssgOLD  3849  prss  3869  tpss  3881  snelpw  4350  sspwb  4354  mss  4364  exss  4365  reg2exmidlema  4679  elomssom  4750  relsn  4878  fnressn  5895  un0mulcl  9580  nn0ssz  9645  hashfibclem  11265  hashf1lem1  11268  hashf1lem2  11269  fimaxre2  11976  fsum2dlemstep  12184  fsumabs  12215  fsumiun  12227  fprod2dlemstep  12372  dvmptfsum  15809  elply2  15819  elplyd  15825  ply1term  15827  plymullem  15834  bdsnss  16882