Theorem sspwb 4702

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 3516	. . . . 5 |- ( x C_ A -> ( A C_ B -> x C_ B ) )
2	1	com12 31	. . . 4 |- ( A C_ B -> ( x C_ A -> x C_ B ) )
3		vex 3121	. . . . 5 |- x e. _V
4	3	elpw 4022	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 4022	. . . 4 |- ( x e. ~P B <-> x C_ B )
6	2, 4, 5	3imtr4g 270	. . 3 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
7	6	ssrdv 3515	. 2 |- ( A C_ B -> ~P A C_ ~P B )
8		ssel 3503	. . . 4 |- ( ~P A C_ ~P B -> ( { x } e. ~P A -> { x } e. ~P B ) )
9		snex 4694	. . . . . 6 |- { x } e. _V
10	9	elpw 4022	. . . . 5 |- ( { x } e. ~P A <-> { x } C_ A )
11	3	snss 4157	. . . . 5 |- ( x e. A <-> { x } C_ A )
12	10, 11	bitr4i 252	. . . 4 |- ( { x } e. ~P A <-> x e. A )
13	9	elpw 4022	. . . . 5 |- ( { x } e. ~P B <-> { x } C_ B )
14	3	snss 4157	. . . . 5 |- ( x e. B <-> { x } C_ B )
15	13, 14	bitr4i 252	. . . 4 |- ( { x } e. ~P B <-> x e. B )
16	8, 12, 15	3imtr3g 269	. . 3 |- ( ~P A C_ ~P B -> ( x e. A -> x e. B ) )
17	16	ssrdv 3515	. 2 |- ( ~P A C_ ~P B -> A C_ B )
18	7, 17	impbii 188	1 |- ( A C_ B <-> ~P A C_ ~P B )

Syntax hints:   <-> wb 184   e. wcel 1767   C_ wss 3481   ~Pcpw 4016   {csn 4033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-sn 4034  df-pr 4036

This theorem is referenced by:  pwel  4705  ssextss  4707  pweqb  4710  pwdom  7681  marypha1lem  7905  wdompwdom  8016  r1pwss  8214  pwwf  8237  rankpwi  8253  rankxplim  8309  ackbij2lem1  8611  fictb  8637  ssfin2  8712  ssfin3ds  8722  ttukeylem2  8902  hashbclem  12482  wrdexg  12538  incexclem  13628  hashbcss  14398  isacs1i  14929  mreacs  14930  acsfn  14931  sscpwex  15062  wunfunc  15143  isacs3lem  15670  isacs5lem  15673  tgcmp  19769  imastopn  20089  fgabs  20248  fgtr  20259  trfg  20260  ssufl  20287  alexsubb  20414  tsmsresOLD  20513  tsmsres  20514  cfiluweak  20666  cfilresi  21602  cmetss  21621  minveclem4a  21713  minveclem4  21715  vitali  21890  sqff1o  23322  elsigagen2  27964  measres  28009  imambfm  28049  ballotlem2  28243  neibastop1  30095  neibastop2lem  30096  neibastop2  30097  sstotbnd2  30188  isnacs3  30561  aomclem2  30920