Theorem int0 3650

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 3255	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 607	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1378	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1629	. . . 4 |- x = x
5	3, 4	2th 172	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2194	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3637	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2603	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2111	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1282   = wceq 1284   e. wcel 1433   {cab 2067   _Vcvv 2601   (/)c0 3251   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-nul 3252  df-int 3637

This theorem is referenced by:  rint0  3675  intexr  3925  bj-intexr  10699