Theorem snss 3754

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3753). 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 3753	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3154  {csn 3619

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167  df-sn 3625

This theorem is referenced by:  snssgOLD  3755  prss  3775  tpss  3785  snelpw  4243  sspwb  4246  mss  4256  exss  4257  reg2exmidlema  4567  elomssom  4638  relsn  4765  fnressn  5745  un0mulcl  9277  nn0ssz  9338  fimaxre2  11373  fsum2dlemstep  11580  fsumabs  11611  fsumiun  11623  fprod2dlemstep  11768  dvmptfsum  14904  elply2  14914  elplyd  14920  ply1term  14922  plymullem  14929  bdsnss  15435