Theorem int0 3860

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 3428	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 646	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1449	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1701	. . . 4 |- x = x
5	3, 4	2th 174	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2293	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3847	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2741	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2208	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1351   = wceq 1353   e. wcel 2148   {cab 2163   _Vcvv 2739   (/)c0 3424   |^|cint 3846

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-nul 3425  df-int 3847

This theorem is referenced by:  rint0  3885  intexr  4152  fiintim  6930  elfi2  6973  fi0  6976  bj-intexr  14745