Theorem uniin 4269

Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 7410 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
uniin

Proof of Theorem uniin

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.40 1679	. . . 4
2		elin 3686	. . . . . . 7
3	2	anbi2i 694	. . . . . 6
4		anandi 828	. . . . . 6
5	3, 4	bitri 249	. . . . 5
6	5	exbii 1667	. . . 4
7		eluni 4252	. . . . 5
8		eluni 4252	. . . . 5
9	7, 8	anbi12i 697	. . . 4
10	1, 6, 9	3imtr4i 266	. . 3
11		eluni 4252	. . 3
12		elin 3686	. . 3
13	10, 11, 12	3imtr4i 266	. 2
14	13	ssriv 3507	1

Syntax hints:  /\wa 369  E.wex 1612  e.wcel 1818  i^icin 3474  C_wss 3475  U.cuni 4249

This theorem is referenced by:  uniinqs  7410  psss  15844  tgval  19456  mapdunirnN  37377

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-uni 4250