Theorem sspwb 3971

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 3006	. . . . 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 2604	. . . . 5 |- x e. _V
4	3	elpw 3388	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3388	. . . 4 |- ( x e. ~P B <-> x C_ B )
6	2, 4, 5	3imtr4g 203	. . 3 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
7	6	ssrdv 3005	. 2 |- ( A C_ B -> ~P A C_ ~P B )
8		ssel 2993	. . . 4 |- ( ~P A C_ ~P B -> ( { x } e. ~P A -> { x } e. ~P B ) )
9	3	snex 3957	. . . . . 6 |- { x } e. _V
10	9	elpw 3388	. . . . 5 |- ( { x } e. ~P A <-> { x } C_ A )
11	3	snss 3516	. . . . 5 |- ( x e. A <-> { x } C_ A )
12	10, 11	bitr4i 185	. . . 4 |- ( { x } e. ~P A <-> x e. A )
13	9	elpw 3388	. . . . 5 |- ( { x } e. ~P B <-> { x } C_ B )
14	3	snss 3516	. . . . 5 |- ( x e. B <-> { x } C_ B )
15	13, 14	bitr4i 185	. . . 4 |- ( { x } e. ~P B <-> x e. B )
16	8, 12, 15	3imtr3g 202	. . 3 |- ( ~P A C_ ~P B -> ( x e. A -> x e. B ) )
17	16	ssrdv 3005	. 2 |- ( ~P A C_ ~P B -> A C_ B )
18	7, 17	impbii 124	1 |- ( A C_ B <-> ~P A C_ ~P B )

Syntax hints:   <-> wb 103   e. wcel 1433   C_ wss 2973   ~Pcpw 3382   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404

This theorem is referenced by:  pwel  3973  ssextss  3975  pweqb  3978