Theorem sspwb 4351

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 3255	. . . . 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 2824	. . . . 5 |- x e. _V
4	3	elpw 3691	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3691	. . . 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 3254	. 2 |- ( A C_ B -> ~P A C_ ~P B )
8		ssel 3242	. . . 4 |- ( ~P A C_ ~P B -> ( { x } e. ~P A -> { x } e. ~P B ) )
9	3	snex 4317	. . . . . 6 |- { x } e. _V
10	9	elpw 3691	. . . . 5 |- ( { x } e. ~P A <-> { x } C_ A )
11	3	snss 3845	. . . . 5 |- ( x e. A <-> { x } C_ A )
12	10, 11	bitr4i 187	. . . 4 |- ( { x } e. ~P A <-> x e. A )
13	9	elpw 3691	. . . . 5 |- ( { x } e. ~P B <-> { x } C_ B )
14	3	snss 3845	. . . . 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 3254	. 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 2209   C_ wss 3220   ~Pcpw 3685   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711

This theorem is referenced by:  pwel  4353  ssextss  4355  pweqb  4358  fiss  7301  pw1on  7575  ntrss  15143