Theorem int0 3979

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 3525	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 655	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1502	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1753	. . . 4 |- x = x
5	3, 4	2th 174	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2354	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3966	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2823	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2269	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   (/)c0 3520   |^|cint 3965

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-nul 3521  df-int 3966

This theorem is referenced by:  rint0  4004  intexr  4281  fiintim  7228  elfi2  7296  fi0  7299  bj-intexr  16848