Theorem intnexr 4241

Description: If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
intnexr	|- ( |^| A = _V -> -. |^| A e. _V )

Proof of Theorem intnexr

Step	Hyp	Ref	Expression
1		vprc 4221	. 2 |- -. _V e. _V
2		eleq1 2294	. 2 |- ( |^| A = _V -> ( |^| A e. _V <-> _V e. _V ) )
3	1, 2	mtbiri 681	1 |- ( |^| A = _V -> -. |^| A e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   |^|cint 3928

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 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2804

This theorem is referenced by: (None)