Theorem sspwb 2750

Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18.

Assertion

Ref	Expression
sspwb	|- (A (_ B <-> P~A (_ P~B)

Proof of Theorem sspwb

Step	Hyp	Ref	Expression
1		sstr2 2067	. . . . 5 |- (x (_ A -> (A (_ B -> x (_ B))
2	1	com12 11	. . . 4 |- (A (_ B -> (x (_ A -> x (_ B))
3		visset 1809	. . . . 5 |- x e. V
4	3	elpw 2400	. . . 4 |- (x e. P~A <-> x (_ A)
5	3	elpw 2400	. . . 4 |- (x e. P~B <-> x (_ B)
6	2, 4, 5	3imtr4g 552	. . 3 |- (A (_ B -> (x e. P~A -> x e. P~B))
7	6	ssrdv 2066	. 2 |- (A (_ B -> P~A (_ P~B)
8		ssel 2059	. . . 4 |- (P~A (_ P~B -> ({x} e. P~A -> {x} e. P~B))
9		snex 2745	. . . . . 6 |- {x} e. V
10	9	elpw 2400	. . . . 5 |- ({x} e. P~A <-> {x} (_ A)
11	3	snss 2457	. . . . 5 |- (x e. A <-> {x} (_ A)
12	10, 11	bitr4 176	. . . 4 |- ({x} e. P~A <-> x e. A)
13	9	elpw 2400	. . . . 5 |- ({x} e. P~B <-> {x} (_ B)
14	3	snss 2457	. . . . 5 |- (x e. B <-> {x} (_ B)
15	13, 14	bitr4 176	. . . 4 |- ({x} e. P~B <-> x e. B)
16	8, 12, 15	3imtr3g 551	. . 3 |- (P~A (_ P~B -> (x e. A -> x e. B))
17	16	ssrdv 2066	. 2 |- (P~A (_ P~B -> A (_ B)
18	7, 17	impbi 157	1 |- (A (_ B <-> P~A (_ P~B)

Syntax hints:   <-> wb 146   e. wcel 956   (_ wss 2043  P~cpw 2397  {csn 2405

This theorem is referenced by:  sspwuni 2753  pwel 2754  ssextss 2757  pweqb 2760  rankpw 4664  rankxplim 4692  fgsb 10480  fgsb2 10485

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408

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