Theorem snss 3806

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3805). 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 3805	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2200  Vcvv 2800   ⊆ wss 3198  {csn 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-in 3204  df-ss 3211  df-sn 3673

This theorem is referenced by:  snssgOLD  3807  prss  3827  tpss  3839  snelpw  4302  sspwb  4306  mss  4316  exss  4317  reg2exmidlema  4630  elomssom  4701  relsn  4829  fnressn  5835  un0mulcl  9429  nn0ssz  9490  fimaxre2  11781  fsum2dlemstep  11988  fsumabs  12019  fsumiun  12031  fprod2dlemstep  12176  dvmptfsum  15442  elply2  15452  elplyd  15458  ply1term  15460  plymullem  15467  bdsnss  16418