Theorem intpr 4320

Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)

Hypotheses

Ref	Expression
intpr.1
intpr.2

Assertion

Ref	Expression
intpr

Proof of Theorem intpr

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

Step	Hyp	Ref	Expression
1		19.26 1680	. . . 4
2		vex 3112	. . . . . . . 8
3	2	elpr 4047	. . . . . . 7
4	3	imbi1i 325	. . . . . 6
5		jaob 783	. . . . . 6
6	4, 5	bitri 249	. . . . 5
7	6	albii 1640	. . . 4
8		intpr.1	. . . . . 6
9	8	clel4 3239	. . . . 5
10		intpr.2	. . . . . 6
11	10	clel4 3239	. . . . 5
12	9, 11	anbi12i 697	. . . 4
13	1, 7, 12	3bitr4i 277	. . 3
14		vex 3112	. . . 4
15	14	elint 4292	. . 3
16		elin 3686	. . 3
17	13, 15, 16	3bitr4i 277	. 2
18	17	eqriv 2453	1

Syntax hints:  ->wi 4  \/wo 368  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  cvv 3109  i^icin 3474  {cpr 4031  |^|cint 4286

This theorem is referenced by:  intprg  4321  uniintsn  4324  op1stb  4722  fiint  7817  shincli  26280  chincli  26378

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-v 3111  df-un 3480  df-in 3482  df-sn 4030  df-pr 4032  df-int 4287