Theorem 0iin 2601

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 2564	. 2 |- |^|_x e. (/) A = {y | A.x e. (/) y e. A}
2		visset 1809	. . . 4 |- y e. V
3		ral0 2354	. . . 4 |- A.x e. (/) y e. A
4	2, 3	2th 717	. . 3 |- (y e. V <-> A.x e. (/) y e. A)
5	4	abbi2i 1571	. 2 |- V = {y | A.x e. (/) y e. A}
6	1, 5	eqtr4 1495	1 |- |^|_x e. (/) A = V

Syntax hints:   = wceq 954   e. wcel 956  {cab 1461  A.wral 1642  Vcvv 1807  (/)c0 2276  |^|_ciin 2562

This theorem is referenced by:  iin0 2735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-dif 2045  df-nul 2277  df-iin 2564

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