Theorem snssg 3845

Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)

Assertion

Ref	Expression
snssg	|- (A e. V -> (A e. B <-> {A} C_ B))

Proof of Theorem snssg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. 2 |- (x = A -> (x e. B <-> A e. B))
2		sneq 3745	. . 3 |- (x = A -> {x} = {A})
3	2	sseq1d 3299	. 2 |- (x = A -> ({x} C_ B <-> {A} C_ B))
4		vex 2863	. . 3 |- x e. _V
5	4	snss 3839	. 2 |- (x e. B <-> {x} C_ B)
6	1, 3, 5	vtoclbg 2916	1 |- (A e. V -> (A e. B <-> {A} C_ B))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710   C_ wss 3258  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-sn 3742

This theorem is referenced by:  snssi  3853  snssd  3854  prssg  3863  snelpwg  4115  elssetkg  4270  nnadjoinpw  4522  spacid  6286