Theorem int0 3785

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 3367	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 635	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1425	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1677	. . . 4 |- x = x
5	3, 4	2th 173	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2255	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3772	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2688	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2170	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1329   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686   (/)c0 3363   |^|cint 3771

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364  df-int 3772

This theorem is referenced by:  rint0  3810  intexr  4075  fiintim  6817  elfi2  6860  fi0  6863  bj-intexr  13106