Theorem pwin 4789

Description: The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)

Assertion

Ref	Expression
pwin

Proof of Theorem pwin

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssin 3719	. . . 4
2		selpw 4019	. . . . 5
3		selpw 4019	. . . . 5
4	2, 3	anbi12i 697	. . . 4
5		selpw 4019	. . . 4
6	1, 4, 5	3bitr4i 277	. . 3
7	6	ineqri 3691	. 2
8	7	eqcomi 2470	1

Syntax hints:  /\wa 369  =wceq 1395  e.wcel 1818  i^icin 3474  C_wss 3475  ~Pcpw 4012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3482  df-ss 3489  df-pw 4014