Theorem int0 3936

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 3495	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 649	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1495	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1747	. . . 4 |- x = x
5	3, 4	2th 174	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2345	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3923	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2801	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2260	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1393   = wceq 1395   e. wcel 2200   {cab 2215   _Vcvv 2799   (/)c0 3491   |^|cint 3922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-nul 3492  df-int 3923

This theorem is referenced by:  rint0  3961  intexr  4233  fiintim  7089  elfi2  7135  fi0  7138  bj-intexr  16229