Theorem prsspw 1858

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class.

Hypotheses

Ref	Expression
prsspw.1	|- A e. V
prsspw.2	|- B e. V

Assertion

Ref	Expression
prsspw	|- ({A, B} (_ P~C <-> (A (_ C /\ B (_ C))

Proof of Theorem prsspw

Step	Hyp	Ref	Expression
1		dfss2 1497	. 2 |- ({A, B} (_ P~C <-> A.x(x e. {A, B} -> x e. P~C))
2		visset 1350	. . . . . 6 |- x e. V
3	2	elpr 1823	. . . . 5 |- (x e. {A, B} <-> (x = A \/ x = B))
4	2	elpw 1801	. . . . 5 |- (x e. P~C <-> x (_ C)
5	3, 4	imbi12i 163	. . . 4 |- ((x e. {A, B} -> x e. P~C) <-> ((x = A \/ x = B) -> x (_ C))
6		jaob 328	. . . 4 |- (((x = A \/ x = B) -> x (_ C) <-> ((x = A -> x (_ C) /\ (x = B -> x (_ C)))
7	5, 6	bitr 151	. . 3 |- ((x e. {A, B} -> x e. P~C) <-> ((x = A -> x (_ C) /\ (x = B -> x (_ C)))
8	7	bial 695	. 2 |- (A.x(x e. {A, B} -> x e. P~C) <-> A.x((x = A -> x (_ C) /\ (x = B -> x (_ C)))
9		19.26 749	. . 3 |- (A.x((x = A -> x (_ C) /\ (x = B -> x (_ C)) <-> (A.x(x = A -> x (_ C) /\ A.x(x = B -> x (_ C)))
10		prsspw.1	. . . . 5 |- A e. V
11		sseq1 1521	. . . . 5 |- (x = A -> (x (_ C <-> A (_ C))
12	10, 11	ceqsalv 1364	. . . 4 |- (A.x(x = A -> x (_ C) <-> A (_ C)
13		prsspw.2	. . . . 5 |- B e. V
14		sseq1 1521	. . . . 5 |- (x = B -> (x (_ C <-> B (_ C))
15	13, 14	ceqsalv 1364	. . . 4 |- (A.x(x = B -> x (_ C) <-> B (_ C)
16	12, 15	anbi12i 369	. . 3 |- ((A.x(x = A -> x (_ C) /\ A.x(x = B -> x (_ C)) <-> (A (_ C /\ B (_ C))
17	9, 16	bitr 151	. 2 |- (A.x((x = A -> x (_ C) /\ (x = B -> x (_ C)) <-> (A (_ C /\ B (_ C))
18	1, 8, 17	3bitr 155	1 |- ({A, B} (_ P~C <-> (A (_ C /\ B (_ C))

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196  A.wal 672   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  P~cpw 1798  {cpr 1809

This theorem is referenced by:  dfchj3 5326

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-pw 1799  df-sn 1811  df-pr 1812

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