Theorem intnexr 4169

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 4150	. 2 ⊢ ¬ V ∈ V
2		eleq1 2252	. 2 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V))
3	1, 2	mtbiri 676	1 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364   ∈ wcel 2160  Vcvv 2752  ∩ cint 3859

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754

This theorem is referenced by: (None)