Theorem 0iin 3818

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 3763	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2644	. . . 4 |- y e. _V
3		ral0 3411	. . . 4 |- A. x e. (/) y e. A
4	2, 3	2th 173	. . 3 |- ( y e. _V <-> A. x e. (/) y e. A )
5	4	abbi2i 2214	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2123	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1299   e. wcel 1448   {cab 2086   A.wral 2375   _Vcvv 2641   (/)c0 3310   |^|_ciin 3761

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-dif 3023  df-nul 3311  df-iin 3763

This theorem is referenced by:  riin0  3831  iin0r  4033