Theorem int0 3846

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)

Assertion

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

Proof of Theorem int0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3419	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 642	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1443	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1695	. . . 4 |- x = x
5	3, 4	2th 173	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2287	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3833	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2733	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2202	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1347   = wceq 1349   e. wcel 2142   {cab 2157   _Vcvv 2731   (/)c0 3415   |^|cint 3832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-dif 3124  df-nul 3416  df-int 3833

This theorem is referenced by:  rint0  3871  intexr  4137  fiintim  6910  elfi2  6953  fi0  6956  bj-intexr  14028