Theorem 0iin 2606

Description: An empty indexed intersection is the universal class.

Assertion

Ref	Expression
0iin	|- |^|_x e. (/) A = V

Proof of Theorem 0iin

Step	Hyp	Ref	Expression
1		df-iin 2569	. 2 |- |^|_x e. (/) A = {y | A.x e. (/) y e. A}
2		visset 1813	. . . 4 |- y e. V
3		ral0 2358	. . . 4 |- A.x e. (/) y e. A
4	2, 3	2th 718	. . 3 |- (y e. V <-> A.x e. (/) y e. A)
5	4	abbi2i 1574	. 2 |- V = {y | A.x e. (/) y e. A}
6	1, 5	eqtr4 1498	1 |- |^|_x e. (/) A = V

Syntax hints:   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  Vcvv 1811  (/)c0 2280  |^|_ciin 2567

This theorem is referenced by:  iin0 2740

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-dif 2049  df-nul 2281  df-iin 2569

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