Theorem snss 3771

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3770). 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 3770	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2177  Vcvv 2773   ⊆ wss 3168  {csn 3635

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-in 3174  df-ss 3181  df-sn 3641

This theorem is referenced by:  snssgOLD  3772  prss  3792  tpss  3802  snelpw  4262  sspwb  4265  mss  4275  exss  4276  reg2exmidlema  4587  elomssom  4658  relsn  4785  fnressn  5780  un0mulcl  9342  nn0ssz  9403  fimaxre2  11588  fsum2dlemstep  11795  fsumabs  11826  fsumiun  11838  fprod2dlemstep  11983  dvmptfsum  15247  elply2  15257  elplyd  15263  ply1term  15265  plymullem  15272  bdsnss  15923