Theorem 0iin 4029

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 3973	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2805	. . . 4 |- y e. _V
3		ral0 3596	. . . 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 1397   e. wcel 2202   {cab 2217   A.wral 2510   _Vcvv 2802   (/)c0 3494   |^|_ciin 3971

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-nul 3495  df-iin 3973

This theorem is referenced by:  riin0  4042  iin0r  4259