Theorem 0iin 4034

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 3978	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2806	. . . 4 |- y e. _V
3		ral0 3598	. . . 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 2346	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2255	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   _Vcvv 2803   (/)c0 3496   |^|_ciin 3976

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-dif 3203  df-nul 3497  df-iin 3978

This theorem is referenced by:  riin0  4047  iin0r  4265