Theorem 0iin 4066

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 4010	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2824	. . . 4 |- y e. _V
3		ral0 3626	. . . 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 2353	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2262	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   _Vcvv 2821   (/)c0 3520   |^|_ciin 4008

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-nul 3521  df-iin 4010

This theorem is referenced by:  riin0  4079  iin0r  4301