Theorem iinxprg 4408

Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)

Hypotheses

Ref	Expression
iinxprg.1
iinxprg.2

Assertion

Ref	Expression
iinxprg

Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem iinxprg

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iinxprg.1	. . . . 5
2	1	eleq2d 2527	. . . 4
3		iinxprg.2	. . . . 5
4	3	eleq2d 2527	. . . 4
5	2, 4	ralprg 4078	. . 3
6	5	abbidv 2593	. 2
7		df-iin 4333	. 2
8		df-in 3482	. 2
9	6, 7, 8	3eqtr4g 2523	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  i^icin 3474  {cpr 4031  |^|_ciin 4331

This theorem is referenced by:  pmapmeet  35497  diameetN  36783  dihmeetlem2N  37026  dihmeetcN  37029  dihmeet  37070

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-3an 975  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-v 3111  df-sbc 3328  df-un 3480  df-in 3482  df-sn 4030  df-pr 4032  df-iin 4333