Theorem intnexr 4203

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 4184	. 2 |- -. _V e. _V
2		eleq1 2269	. 2 |- ( |^| A = _V -> ( |^| A e. _V <-> _V e. _V ) )
3	1, 2	mtbiri 677	1 |- ( |^| A = _V -> -. |^| A e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1373   e. wcel 2177   _Vcvv 2773   |^|cint 3891

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 615  ax-in2 616  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-v 2775

This theorem is referenced by: (None)