Theorem snssg 1850

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 1149	. 2 |- (x = A -> (x e. B <-> A e. B))
2		sneq 1816	. . 3 |- (x = A -> {x} = {A})
3	2	sseq1d 1527	. 2 |- (x = A -> ({x} (_ B <-> {A} (_ B))
4		visset 1350	. . 3 |- x e. V
5	4	snss 1849	. 2 |- (x e. B <-> {x} (_ B)
6	1, 3, 5	vtoclbg 1384	1 |- (A e. C -> (A e. B <-> {A} (_ B))

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092   (_ wss 1487  {csn 1808

This theorem is referenced by:  snssi 1851  h1did 5456

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-sn 1811

metamath.org