Theorem int0 3838

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 3413	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 636	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1437	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1689	. . . 4 |- x = x
5	3, 4	2th 173	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2282	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3825	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2728	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2196	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   e. wcel 2136   {cab 2151   _Vcvv 2726   (/)c0 3409   |^|cint 3824

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-nul 3410  df-int 3825

This theorem is referenced by:  rint0  3863  intexr  4129  fiintim  6894  elfi2  6937  fi0  6940  bj-intexr  13790