Theorem snssg 2467

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 1537	. 2 |- (x = A -> (x e. B <-> A e. B))
2		sneq 2421	. . 3 |- (x = A -> {x} = {A})
3	2	sseq1d 2091	. 2 |- (x = A -> ({x} (_ B <-> {A} (_ B))
4		visset 1816	. . 3 |- x e. V
5	4	snss 2465	. 2 |- (x e. B <-> {x} (_ B)
6	1, 3, 5	vtoclbg 1851	1 |- (A e. C -> (A e. B <-> {A} (_ B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960   (_ wss 2050  {csn 2413

This theorem is referenced by:  snssi 2470  fvimacnvALT 3815  isneip 7717  elnei 7722  h1did 9469  cnfilca 10562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056  df-sn 2416

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