Theorem intprd 10461

Description: The intersection of a pair is the intersection of its members. Closed for of intpr 2567.Theorem 71 of [Suppes] p. 42.

Assertion

Ref	Expression
intprd	|- ((A e. V /\ B e. V) -> |^|{A, B} = (A i^i B))

Proof of Theorem intprd

Step	Hyp	Ref	Expression
1		preq1 2452	. . . 4 |- (A = if(A e. V, A, (/)) -> {A, B} = {if(A e. V, A, (/)), B})
2	1	inteqd 2542	. . 3 |- (A = if(A e. V, A, (/)) -> |^|{A, B} = |^|{if(A e. V, A, (/)), B})
3		ineq1 2213	. . 3 |- (A = if(A e. V, A, (/)) -> (A i^i B) = (if(A e. V, A, (/)) i^i B))
4	2, 3	eqeq12d 1492	. 2 |- (A = if(A e. V, A, (/)) -> (|^|{A, B} = (A i^i B) <-> |^|{if(A e. V, A, (/)), B} = (if(A e. V, A, (/)) i^i B)))
5		preq2 2453	. . . 4 |- (B = if(B e. V, B, (/)) -> {if(A e. V, A, (/)), B} = {if(A e. V, A, (/)), if(B e. V, B, (/))})
6	5	inteqd 2542	. . 3 |- (B = if(B e. V, B, (/)) -> |^|{if(A e. V, A, (/)), B} = |^|{if(A e. V, A, (/)), if(B e. V, B, (/))})
7		ineq2 2214	. . 3 |- (B = if(B e. V, B, (/)) -> (if(A e. V, A, (/)) i^i B) = (if(A e. V, A, (/)) i^i if(B e. V, B, (/))))
8	6, 7	eqeq12d 1492	. 2 |- (B = if(B e. V, B, (/)) -> (|^|{if(A e. V, A, (/)), B} = (if(A e. V, A, (/)) i^i B) <-> |^|{if(A e. V, A, (/)), if(B e. V, B, (/))} = (if(A e. V, A, (/)) i^i if(B e. V, B, (/)))))
9		0ex 2716	. . . 4 |- (/) e. V
10	9	elimel 2398	. . 3 |- if(A e. V, A, (/)) e. V
11	9	elimel 2398	. . 3 |- if(B e. V, B, (/)) e. V
12	10, 11	intpr 2567	. 2 |- |^|{if(A e. V, A, (/)), if(B e. V, B, (/))} = (if(A e. V, A, (/)) i^i if(B e. V, B, (/)))
13	4, 8, 12	dedth2h 2391	1 |- ((A e. V /\ B e. V) -> |^|{A, B} = (A i^i B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814   i^i cin 2049  (/)c0 2283  ifcif 2365  {cpr 2414  |^|cint 2537

This theorem is referenced by:  cnfilca 10562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-nul 2715

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-nul 2284  df-if 2366  df-sn 2416  df-pr 2417  df-int 2538

Copyright terms: Public domain