Theorem bj-intnexr 15565

Description: intnexr 4185 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-intnexr	⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)

Proof of Theorem bj-intnexr

Step	Hyp	Ref	Expression
1		bj-vprc 15552	. 2 ⊢ ¬ V ∈ V
2		eleq1 2259	. 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 2167  Vcvv 2763  ∩ cint 3875

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-13 2169  ax-14 2170  ax-ext 2178  ax-bdn 15473  ax-bdel 15477  ax-bdsep 15540

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by: (None)