Theorem 0iin 4578

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 4523	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 3203	. . . 4 |- y e. _V
3		ral0 4076	. . . 4 |- A. x e. (/) y e. A
4	2, 3	2th 254	. . 3 |- ( y e. _V <-> A. x e. (/) y e. A )
5	4	abbi2i 2738	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2647	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   (/)c0 3915   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916  df-iin 4523

This theorem is referenced by:  iinrab2  4583  iinvdif  4592  riin0  4594  iin0  4839  xpriindi  5258  cmpfi  21211  ptbasfi  21384  pol0N  35195