Theorem snsspw 2476

Description: The singleton of a class is a subset of its power class.

Assertion

Ref	Expression
snsspw	|- {A} (_ P~A

Proof of Theorem snsspw

Step	Hyp	Ref	Expression
1		eqimss 2106	. . 3 |- (x = A -> x (_ A)
2		elsn 2418	. . 3 |- (x e. {A} <-> x = A)
3		df-pw 2399	. . . 4 |- P~A = {x | x (_ A}
4	3	abeq2i 1568	. . 3 |- (x e. P~A <-> x (_ A)
5	1, 2, 4	3imtr4 219	. 2 |- (x e. {A} -> x e. P~A)
6	5	ssriv 2066	1 |- {A} (_ P~A

Syntax hints:   = wceq 955   e. wcel 957   (_ wss 2044  P~cpw 2398  {csn 2406

This theorem is referenced by:  snex 2746

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-in 2048  df-ss 2050  df-pw 2399  df-sn 2409

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