Theorem iundif2 4397

Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4384 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Assertion

Ref	Expression
iundif2

Distinct variable group:   ,

Proof of Theorem iundif2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3485	. . . . 5
2	1	rexbii 2959	. . . 4
3		r19.42v 3012	. . . 4
4		rexnal 2905	. . . . . 6
5		vex 3112	. . . . . . 7
6		eliin 4336	. . . . . . 7
7	5, 6	ax-mp 5	. . . . . 6
8	4, 7	xchbinxr 311	. . . . 5
9	8	anbi2i 694	. . . 4
10	2, 3, 9	3bitri 271	. . 3
11		eliun 4335	. . 3
12		eldif 3485	. . 3
13	10, 11, 12	3bitr4i 277	. 2
14	13	eqriv 2453	1

Syntax hints:  -.wn 3  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109  \cdif 3472  U_ciun 4330  |^|_ciin 4331

This theorem is referenced by:  iuncld  19546  pnrmopn  19844  alexsublem  20544  bcth3  21770  iundifdifd  27429  iundifdif  27430

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-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-iun 4332  df-iin 4333