Theorem snsspw 3878

Description: The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
snsspw	|- {A} C_ ~PA

Proof of Theorem snsspw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqimss 3324	. . 3 |- (x = A -> x C_ A)
2		elsn 3749	. . 3 |- (x e. {A} <-> x = A)
3		df-pw 3725	. . . 4 |- ~PA = {x | x C_ A}
4	3	abeq2i 2461	. . 3 |- (x e. ~PA <-> x C_ A)
5	1, 2, 4	3imtr4i 257	. 2 |- (x e. {A} -> x e. ~PA)
6	5	ssriv 3278	1 |- {A} C_ ~PA

Syntax hints:   = wceq 1642   e. wcel 1710   C_ wss 3258  ~Pcpw 3723  {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-pw 3725  df-sn 3742

This theorem is referenced by: (None)