Theorem int0 3629

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 3228	. . . . . 6 |- -. y e. (/)
2	1	pm2.21i 575	. . . . 5 |- ( y e. (/) -> x e. y )
3	2	ax-gen 1338	. . . 4 |- A. y ( y e. (/) -> x e. y )
4		equid 1589	. . . 4 |- x = x
5	3, 4	2th 163	. . 3 |- ( A. y ( y e. (/) -> x e. y ) <-> x = x )
6	5	abbii 2153	. 2 |- { x | A. y ( y e. (/) -> x e. y ) } = { x | x = x }
7		df-int 3616	. 2 |- |^| (/) = { x | A. y ( y e. (/) -> x e. y ) }
8		df-v 2559	. 2 |- _V = { x | x = x }
9	6, 7, 8	3eqtr4i 2070	1 |- |^| (/) = _V

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   _Vcvv 2557   (/)c0 3224   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225  df-int 3616

This theorem is referenced by:  rint0  3654  intexr  3904  bj-intexr  10028