Theorem sspwb 4210

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 3160	. . . . 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 2738	. . . . 5 |- x e. _V
4	3	elpw 3578	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3578	. . . 4 |- ( x e. ~P B <-> x C_ B )
6	2, 4, 5	3imtr4g 205	. . 3 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
7	6	ssrdv 3159	. 2 |- ( A C_ B -> ~P A C_ ~P B )
8		ssel 3147	. . . 4 |- ( ~P A C_ ~P B -> ( { x } e. ~P A -> { x } e. ~P B ) )
9	3	snex 4180	. . . . . 6 |- { x } e. _V
10	9	elpw 3578	. . . . 5 |- ( { x } e. ~P A <-> { x } C_ A )
11	3	snss 3724	. . . . 5 |- ( x e. A <-> { x } C_ A )
12	10, 11	bitr4i 187	. . . 4 |- ( { x } e. ~P A <-> x e. A )
13	9	elpw 3578	. . . . 5 |- ( { x } e. ~P B <-> { x } C_ B )
14	3	snss 3724	. . . . 5 |- ( x e. B <-> { x } C_ B )
15	13, 14	bitr4i 187	. . . 4 |- ( { x } e. ~P B <-> x e. B )
16	8, 12, 15	3imtr3g 204	. . 3 |- ( ~P A C_ ~P B -> ( x e. A -> x e. B ) )
17	16	ssrdv 3159	. 2 |- ( ~P A C_ ~P B -> A C_ B )
18	7, 17	impbii 126	1 |- ( A C_ B <-> ~P A C_ ~P B )

Syntax hints:   <-> wb 105   e. wcel 2146   C_ wss 3127   ~Pcpw 3572   {csn 3589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595

This theorem is referenced by:  pwel  4212  ssextss  4214  pweqb  4217  fiss  6966  pw1on  7215  ntrss  13190