Theorem sspwb 3943

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 <-> ~PA C_ ~PB)

Proof of Theorem sspwb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 2946	. . . . 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 2554	. . . . 5 |- x e. _V
4	3	elpw 3357	. . . 4 |- (x e. ~PA <-> x C_ A)
5	3	elpw 3357	. . . 4 |- (x e. ~PB <-> x C_ B)
6	2, 4, 5	3imtr4g 194	. . 3 |- (A C_ B -> (x e. ~PA -> x e. ~PB))
7	6	ssrdv 2945	. 2 |- (A C_ B -> ~PA C_ ~PB)
8		ssel 2933	. . . 4 |- ( ~PA C_ ~PB -> ( {x } e. ~PA -> {x } e. ~PB))
9		snexgOLD 3926	. . . . . . 7 |- (x e. _V -> {x } e. _V)
10	3, 9	ax-mp 7	. . . . . 6 |- {x } e. _V
11	10	elpw 3357	. . . . 5 |- ( {x } e. ~PA <-> {x } C_ A)
12	3	snss 3485	. . . . 5 |- (x e. A <-> {x } C_ A)
13	11, 12	bitr4i 176	. . . 4 |- ( {x } e. ~PA <-> x e. A)
14	10	elpw 3357	. . . . 5 |- ( {x } e. ~PB <-> {x } C_ B)
15	3	snss 3485	. . . . 5 |- (x e. B <-> {x } C_ B)
16	14, 15	bitr4i 176	. . . 4 |- ( {x } e. ~PB <-> x e. B)
17	8, 13, 16	3imtr3g 193	. . 3 |- ( ~PA C_ ~PB -> (x e. A -> x e. B))
18	17	ssrdv 2945	. 2 |- ( ~PA C_ ~PB -> A C_ B)
19	7, 18	impbii 117	1 |- (A C_ B <-> ~PA C_ ~PB)

Syntax hints:   <-> wb 98   e. wcel 1390   _Vcvv 2551   C_ wss 2911   ~Pcpw 3351   {csn 3367

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by:  pwel  3945  ssextss  3947  pweqb  3950