Theorem iindif2 4399

Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4383 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)

Assertion

Ref	Expression
iindif2

Distinct variable groups:   ,   ,

Proof of Theorem iindif2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28zv 3924	. . . 4
2		eldif 3485	. . . . . 6
3	2	bicomi 202	. . . . 5
4	3	ralbii 2888	. . . 4
5		ralnex 2903	. . . . . 6
6		eliun 4335	. . . . . 6
7	5, 6	xchbinxr 311	. . . . 5
8	7	anbi2i 694	. . . 4
9	1, 4, 8	3bitr3g 287	. . 3
10		vex 3112	. . . 4
11		eliin 4336	. . . 4
12	10, 11	ax-mp 5	. . 3
13		eldif 3485	. . 3
14	9, 12, 13	3bitr4g 288	. 2
15	14	eqrdv 2454	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  cvv 3109  \cdif 3472  c0 3784  U_ciun 4330  |^|_ciin 4331

This theorem is referenced by:  iinvdif  4402  iincld  19540  clsval2  19551  mretopd  19593  hauscmplem  19906  cmpfi  19908

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-or 370  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-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-nul 3785  df-iun 4332  df-iin 4333