Theorem intprg 4321

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4320. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

Assertion

Ref	Expression
intprg

Proof of Theorem intprg

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

Step	Hyp	Ref	Expression
1		preq1 4109	. . . 4
2	1	inteqd 4291	. . 3
3		ineq1 3692	. . 3
4	2, 3	eqeq12d 2479	. 2
5		preq2 4110	. . . 4
6	5	inteqd 4291	. . 3
7		ineq2 3693	. . 3
8	6, 7	eqeq12d 2479	. 2
9		vex 3112	. . 3
10		vex 3112	. . 3
11	9, 10	intpr 4320	. 2
12	4, 8, 11	vtocl2g 3171	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  i^icin 3474  {cpr 4031  |^|cint 4286

This theorem is referenced by:  intsng  4322  inelfi  7898  mreincl  14996  subrgin  17452  lssincl  17611  incld  19544  difelsiga  28133  inidl  30427

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-un 3480  df-in 3482  df-sn 4030  df-pr 4032  df-int 4287