Theorem sspwb 4261

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 3200	. . . . 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 2775	. . . . 5 |- x e. _V
4	3	elpw 3622	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3622	. . . 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 3199	. 2 |- ( A C_ B -> ~P A C_ ~P B )
8		ssel 3187	. . . 4 |- ( ~P A C_ ~P B -> ( { x } e. ~P A -> { x } e. ~P B ) )
9	3	snex 4230	. . . . . 6 |- { x } e. _V
10	9	elpw 3622	. . . . 5 |- ( { x } e. ~P A <-> { x } C_ A )
11	3	snss 3768	. . . . 5 |- ( x e. A <-> { x } C_ A )
12	10, 11	bitr4i 187	. . . 4 |- ( { x } e. ~P A <-> x e. A )
13	9	elpw 3622	. . . . 5 |- ( { x } e. ~P B <-> { x } C_ B )
14	3	snss 3768	. . . . 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 3199	. 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 2176   C_ wss 3166   ~Pcpw 3616   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639

This theorem is referenced by:  pwel  4263  ssextss  4265  pweqb  4268  fiss  7081  pw1on  7340  ntrss  14624