Theorem 0iin 3736

Description: An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)

Assertion

Ref	Expression
0iin	|- |^|_ x e. (/) A = _V

Proof of Theorem 0iin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3681	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2604	. . . 4 |- y e. _V
3		ral0 3342	. . . 4 |- A. x e. (/) y e. A
4	2, 3	2th 172	. . 3 |- ( y e. _V <-> A. x e. (/) y e. A )
5	4	abbi2i 2193	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2104	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   _Vcvv 2601   (/)c0 3251   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-dif 2975  df-nul 3252  df-iin 3681

This theorem is referenced by:  riin0  3749  iin0r  3943