Theorem 0iin 4000

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 3944	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2779	. . . 4 |- y e. _V
3		ral0 3570	. . . 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 2322	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2231	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   _Vcvv 2776   (/)c0 3468   |^|_ciin 3942

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-dif 3176  df-nul 3469  df-iin 3944

This theorem is referenced by:  riin0  4013  iin0r  4229