Theorem int0 3708

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 3291	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 611	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1384	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1635	. . . 4 |- x = x
5	3, 4	2th 173	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2204	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3695	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2622	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2119	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1288   = wceq 1290   e. wcel 1439   {cab 2075   _Vcvv 2620   (/)c0 3287   |^|cint 3694

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-nul 3288  df-int 3695

This theorem is referenced by:  rint0  3733  intexr  3992  fiintim  6693  bj-intexr  12072