Theorem sspwb 3952

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 2952	. . . . 5 |- ( x C_ A -> ( A C_ B -> x C_ B ) )
2	1	com12 27	. . . 4 |- ( A C_ B -> ( x C_ A -> x C_ B ) )
3		vex 2560	. . . . 5 |- x e. _V
4	3	elpw 3365	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3365	. . . 4 |- ( x e. ~P B <-> x C_ B )
6	2, 4, 5	3imtr4g 194	. . 3 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
7	6	ssrdv 2951	. 2 |- ( A C_ B -> ~P A C_ ~P B )
8		ssel 2939	. . . 4 |- ( ~P A C_ ~P B -> ( { x } e. ~P A -> { x } e. ~P B ) )
9		snexgOLD 3935	. . . . . . 7 |- ( x e. _V -> { x } e. _V )
10	3, 9	ax-mp 7	. . . . . 6 |- { x } e. _V
11	10	elpw 3365	. . . . 5 |- ( { x } e. ~P A <-> { x } C_ A )
12	3	snss 3494	. . . . 5 |- ( x e. A <-> { x } C_ A )
13	11, 12	bitr4i 176	. . . 4 |- ( { x } e. ~P A <-> x e. A )
14	10	elpw 3365	. . . . 5 |- ( { x } e. ~P B <-> { x } C_ B )
15	3	snss 3494	. . . . 5 |- ( x e. B <-> { x } C_ B )
16	14, 15	bitr4i 176	. . . 4 |- ( { x } e. ~P B <-> x e. B )
17	8, 13, 16	3imtr3g 193	. . 3 |- ( ~P A C_ ~P B -> ( x e. A -> x e. B ) )
18	17	ssrdv 2951	. 2 |- ( ~P A C_ ~P B -> A C_ B )
19	7, 18	impbii 117	1 |- ( A C_ B <-> ~P A C_ ~P B )

Syntax hints:   <-> wb 98   e. wcel 1393   _Vcvv 2557   C_ wss 2917   ~Pcpw 3359   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by:  pwel  3954  ssextss  3956  pweqb  3959