Theorem snss 3803

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3802). 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 3802	. 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 2799   ⊆ wss 3197  {csn 3666

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 2801  df-in 3203  df-ss 3210  df-sn 3672

This theorem is referenced by:  snssgOLD  3804  prss  3824  tpss  3836  snelpw  4298  sspwb  4302  mss  4312  exss  4313  reg2exmidlema  4626  elomssom  4697  relsn  4824  fnressn  5829  un0mulcl  9411  nn0ssz  9472  fimaxre2  11746  fsum2dlemstep  11953  fsumabs  11984  fsumiun  11996  fprod2dlemstep  12141  dvmptfsum  15407  elply2  15417  elplyd  15423  ply1term  15425  plymullem  15432  bdsnss  16260