Theorem snssg 2463

Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49.

Assertion

Ref	Expression
snssg	|- (A e. C -> (A e. B <-> {A} (_ B))

Proof of Theorem snssg

Step	Hyp	Ref	Expression
1		eleq1 1534	. 2 |- (x = A -> (x e. B <-> A e. B))
2		sneq 2417	. . 3 |- (x = A -> {x} = {A})
3	2	sseq1d 2088	. 2 |- (x = A -> ({x} (_ B <-> {A} (_ B))
4		visset 1813	. . 3 |- x e. V
5	4	snss 2461	. 2 |- (x e. B <-> {x} (_ B)
6	1, 3, 5	vtoclbg 1848	1 |- (A e. C -> (A e. B <-> {A} (_ B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958   (_ wss 2047  {csn 2409

This theorem is referenced by:  snssi 2466  fvimacnvALT 3809  isneip 7720  elnei 7725  h1did 9474  cnfilca 10583  cnfilcaOLD 10584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-sn 2412

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