Theorem 0iin 3971

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 3915	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2763	. . . 4 |- y e. _V
3		ral0 3548	. . . 4 |- A. x e. (/) y e. A
4	2, 3	2th 174	. . 3 |- ( y e. _V <-> A. x e. (/) y e. A )
5	4	abbi2i 2308	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2217	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   _Vcvv 2760   (/)c0 3446   |^|_ciin 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-dif 3155  df-nul 3447  df-iin 3915

This theorem is referenced by:  riin0  3984  iin0r  4198