Theorem sspwb 4201

Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)

Assertion

Ref	Expression
sspwb	|- ( A C_ B <-> ~P A C_ ~P B )

Proof of Theorem sspwb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3154	. . . . 5 |- ( x C_ A -> ( A C_ B -> x C_ B ) )
2	1	com12 30	. . . 4 |- ( A C_ B -> ( x C_ A -> x C_ B ) )
3		vex 2733	. . . . 5 |- x e. _V
4	3	elpw 3572	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3572	. . . 4 |- ( x e. ~P B <-> x C_ B )
6	2, 4, 5	3imtr4g 204	. . 3 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
7	6	ssrdv 3153	. 2 |- ( A C_ B -> ~P A C_ ~P B )
8		ssel 3141	. . . 4 |- ( ~P A C_ ~P B -> ( { x } e. ~P A -> { x } e. ~P B ) )
9	3	snex 4171	. . . . . 6 |- { x } e. _V
10	9	elpw 3572	. . . . 5 |- ( { x } e. ~P A <-> { x } C_ A )
11	3	snss 3709	. . . . 5 |- ( x e. A <-> { x } C_ A )
12	10, 11	bitr4i 186	. . . 4 |- ( { x } e. ~P A <-> x e. A )
13	9	elpw 3572	. . . . 5 |- ( { x } e. ~P B <-> { x } C_ B )
14	3	snss 3709	. . . . 5 |- ( x e. B <-> { x } C_ B )
15	13, 14	bitr4i 186	. . . 4 |- ( { x } e. ~P B <-> x e. B )
16	8, 12, 15	3imtr3g 203	. . 3 |- ( ~P A C_ ~P B -> ( x e. A -> x e. B ) )
17	16	ssrdv 3153	. 2 |- ( ~P A C_ ~P B -> A C_ B )
18	7, 17	impbii 125	1 |- ( A C_ B <-> ~P A C_ ~P B )

Syntax hints:   <-> wb 104   e. wcel 2141   C_ wss 3121   ~Pcpw 3566   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by:  pwel  4203  ssextss  4205  pweqb  4208  fiss  6954  pw1on  7203  ntrss  12913