Theorem intnexr 4146

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 4130	. 2 |- -. _V e. _V
2		eleq1 2238	. 2 |- ( |^| A = _V -> ( |^| A e. _V <-> _V e. _V ) )
3	1, 2	mtbiri 675	1 |- ( |^| A = _V -> -. |^| A e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   e. wcel 2146   _Vcvv 2735   |^|cint 3840

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-5 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-v 2737

This theorem is referenced by: (None)