Theorem 0iin 3715

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 3660	. 2 |- |^|_ x e. (/) A = { y | A. x e. (/) y e. A }
2		vex 2560	. . . 4 |- y e. _V
3		ral0 3322	. . . 4 |- A. x e. (/) y e. A
4	2, 3	2th 163	. . 3 |- ( y e. _V <-> A. x e. (/) y e. A )
5	4	abbi2i 2152	. 2 |- _V = { y | A. x e. (/) y e. A }
6	1, 5	eqtr4i 2063	1 |- |^|_ x e. (/) A = _V

Syntax hints:   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   _Vcvv 2557   (/)c0 3224   |^|_ciin 3658

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-nul 3225  df-iin 3660

This theorem is referenced by:  riin0  3728  iin0r  3922