Theorem int0 2515

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 2255	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 77	. . . . 5 |- (y e. (/) -> x e. y)
3	2	ax-gen 955	. . . 4 |- A.y(y e. (/) -> x e. y)
4		eqid 1452	. . . 4 |- x = x
5	3, 4	2th 715	. . 3 |- (A.y(y e. (/) -> x e. y) <-> x = x)
6	5	abbii 1551	. 2 |- {x | A.y(y e. (/) -> x e. y)} = {x | x = x}
7		df-int 2502	. 2 |- |^|(/) = {x | A.y(y e. (/) -> x e. y)}
8		df-v 1787	. 2 |- V = {x | x = x}
9	6, 7, 8	3eqtr4 1481	1 |- |^|(/) = V

Syntax hints:   -> wi 3  A.wal 950   = wceq 1099   e. wcel 1105  {cab 1440  Vcvv 1786  (/)c0 2251  |^|cint 2501

This theorem is referenced by:  intex 2697  intnex 2698  oev2 4100  fiint 4486  fiiu 8709  fiiu2 8734  efilcp 8795  efilcp2 8800  cnfilca 8801

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-dif 2020  df-nul 2252  df-int 2502

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