Theorem int0 3898

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 3463	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 647	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1471	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1723	. . . 4 |- x = x
5	3, 4	2th 174	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2320	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3885	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2773	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2235	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1370   = wceq 1372   e. wcel 2175   {cab 2190   _Vcvv 2771   (/)c0 3459   |^|cint 3884

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-nul 3460  df-int 3885

This theorem is referenced by:  rint0  3923  intexr  4193  fiintim  7010  elfi2  7056  fi0  7059  bj-intexr  15708