Theorem int0 3888

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 3454	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 647	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1463	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1715	. . . 4 |- x = x
5	3, 4	2th 174	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2312	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3875	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2765	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2227	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1362   = wceq 1364   e. wcel 2167   {cab 2182   _Vcvv 2763   (/)c0 3450   |^|cint 3874

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-nul 3451  df-int 3875

This theorem is referenced by:  rint0  3913  intexr  4183  fiintim  6992  elfi2  7038  fi0  7041  bj-intexr  15554