Theorem prsspw 2471

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 2048	. 2 |- ({A, B} (_ P~C <-> A.x(x e. {A, B} -> x e. P~C))
2		visset 1804	. . . . . 6 |- x e. V
3	2	elpr 2414	. . . . 5 |- (x e. {A, B} <-> (x = A \/ x = B))
4	2	elpw 2394	. . . . 5 |- (x e. P~C <-> x (_ C)
5	3, 4	imbi12i 188	. . . 4 |- ((x e. {A, B} -> x e. P~C) <-> ((x = A \/ x = B) -> x (_ C))
6		jaob 422	. . . 4 |- (((x = A \/ x = B) -> x (_ C) <-> ((x = A -> x (_ C) /\ (x = B -> x (_ C)))
7	5, 6	bitr 173	. . 3 |- ((x e. {A, B} -> x e. P~C) <-> ((x = A -> x (_ C) /\ (x = B -> x (_ C)))
8	7	albii 996	. 2 |- (A.x(x e. {A, B} -> x e. P~C) <-> A.x((x = A -> x (_ C) /\ (x = B -> x (_ C)))
9		19.26 1063	. . 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 2072	. . . . 5 |- (x = A -> (x (_ C <-> A (_ C))
12	10, 11	ceqsalv 1818	. . . 4 |- (A.x(x = A -> x (_ C) <-> A (_ C)
13		prsspw.2	. . . . 5 |- B e. V
14		sseq1 2072	. . . . 5 |- (x = B -> (x (_ C <-> B (_ C))
15	13, 14	ceqsalv 1818	. . . 4 |- (A.x(x = B -> x (_ C) <-> B (_ C)
16	12, 15	anbi12i 481	. . 3 |- ((A.x(x = A -> x (_ C) /\ A.x(x = B -> x (_ C)) <-> (A (_ C /\ B (_ C))
17	9, 16	bitr 173	. 2 |- (A.x((x = A -> x (_ C) /\ (x = B -> x (_ C)) <-> (A (_ C /\ B (_ C))
18	1, 8, 17	3bitr 177	1 |- ({A, B} (_ P~C <-> (A (_ C /\ B (_ C))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  Vcvv 1802   (_ wss 2037  P~cpw 2391  {cpr 2400

This theorem is referenced by:  dfchj3 9240

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-in 2041  df-ss 2043  df-pw 2392  df-sn 2402  df-pr 2403

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