Theorem int0 2543

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44.

Assertion

Ref	Expression
int0	|- |^|(/) = V

Proof of Theorem int0

Step	Hyp	Ref	Expression
1		noel 2281	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 77	. . . . 5 |- (y e. (/) -> x e. y)
3	2	ax-gen 962	. . . 4 |- A.y(y e. (/) -> x e. y)
4		eqid 1474	. . . 4 |- x = x
5	3, 4	2th 717	. . 3 |- (A.y(y e. (/) -> x e. y) <-> x = x)
6	5	abbii 1573	. 2 |- {x | A.y(y e. (/) -> x e. y)} = {x | x = x}
7		df-int 2530	. 2 |- |^|(/) = {x | A.y(y e. (/) -> x e. y)}
8		df-v 1809	. 2 |- V = {x | x = x}
9	6, 7, 8	3eqtr4 1503	1 |- |^|(/) = V

Syntax hints:   -> wi 3  A.wal 953   = wceq 955   e. wcel 957  {cab 1462  Vcvv 1808  (/)c0 2277  |^|cint 2529

This theorem is referenced by:  intex 2725  intnex 2726  oev2 4155  fiint 4543  fiiu 10408  fiiu2 10436  efilcp 10504  efilcp2 10509  cnfilca 10510

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-dif 2046  df-nul 2278  df-int 2530

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