Theorem 0iin 3931

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 3876	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2733	. . . 4 |- y e. _V
3		ral0 3516	. . . 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 2285	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2194	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   _Vcvv 2730   (/)c0 3414   |^|_ciin 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-nul 3415  df-iin 3876

This theorem is referenced by:  riin0  3944  iin0r  4155