Theorem int0 3793

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 3372	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 636	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1426	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1678	. . . 4 |- x = x
5	3, 4	2th 173	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2256	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3780	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2691	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2171	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1330   = wceq 1332   e. wcel 1481   {cab 2126   _Vcvv 2689   (/)c0 3368   |^|cint 3779

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-nul 3369  df-int 3780

This theorem is referenced by:  rint0  3818  intexr  4083  fiintim  6825  elfi2  6868  fi0  6871  bj-intexr  13277