Theorem intnexr 4138

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 4122	. 2 |- -. _V e. _V
2		eleq1 2234	. 2 |- ( |^| A = _V -> ( |^| A e. _V <-> _V e. _V ) )
3	1, 2	mtbiri 671	1 |- ( |^| A = _V -> -. |^| A e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1349   e. wcel 2142   _Vcvv 2731   |^|cint 3832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-5 1441  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-13 2144  ax-14 2145  ax-ext 2153  ax-sep 4108

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-v 2733

This theorem is referenced by: (None)