Theorem int0 3947

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 3500	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 651	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1498	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1749	. . . 4 |- x = x
5	3, 4	2th 174	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2347	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3934	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2805	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2262	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1396   = wceq 1398   e. wcel 2202   {cab 2217   _Vcvv 2803   (/)c0 3496   |^|cint 3933

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-nul 3497  df-int 3934

This theorem is referenced by:  rint0  3972  intexr  4245  fiintim  7166  elfi2  7214  fi0  7217  bj-intexr  16607