Theorem int0 3697

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 3288	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 610	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1383	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1634	. . . 4 |- x = x
5	3, 4	2th 172	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2203	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3684	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2621	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2118	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1287   = wceq 1289   e. wcel 1438   {cab 2074   _Vcvv 2619   (/)c0 3284   |^|cint 3683

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-nul 3285  df-int 3684

This theorem is referenced by:  rint0  3722  intexr  3978  bj-intexr  11456