Theorem int0 3905

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 3468	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 647	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1473	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1725	. . . 4 |- x = x
5	3, 4	2th 174	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2322	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3892	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2775	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2237	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1371   = wceq 1373   e. wcel 2177   {cab 2192   _Vcvv 2773   (/)c0 3464   |^|cint 3891

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-nul 3465  df-int 3892

This theorem is referenced by:  rint0  3930  intexr  4202  fiintim  7043  elfi2  7089  fi0  7092  bj-intexr  15982