Theorem intnexr 3985

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 ∈ V)

Proof of Theorem intnexr

Step	Hyp	Ref	Expression
1		vprc 3969	. 2 ⊢ ¬ V ∈ V
2		eleq1 2150	. 2 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V))
3	1, 2	mtbiri 635	1 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1289   ∈ wcel 1438  Vcvv 2619  ∩ cint 3686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070  ax-sep 3955

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621

This theorem is referenced by: (None)