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Theorem intnex 5344
Description: If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
Assertion
Ref Expression
intnex 𝐴 ∈ V ↔ 𝐴 = V)

Proof of Theorem intnex
StepHypRef Expression
1 intex 5343 . . . 4 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
21necon1bbii 2989 . . 3 𝐴 ∈ V ↔ 𝐴 = ∅)
3 inteq 4948 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
4 int0 4961 . . . 4 ∅ = V
53, 4eqtrdi 2792 . . 3 (𝐴 = ∅ → 𝐴 = V)
62, 5sylbi 217 . 2 𝐴 ∈ V → 𝐴 = V)
7 vprc 5314 . . 3 ¬ V ∈ V
8 eleq1 2828 . . 3 ( 𝐴 = V → ( 𝐴 ∈ V ↔ V ∈ V))
97, 8mtbiri 327 . 2 ( 𝐴 = V → ¬ 𝐴 ∈ V)
106, 9impbii 209 1 𝐴 ∈ V ↔ 𝐴 = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1539  wcel 2107  Vcvv 3479  c0 4332   cint 4945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967  df-nul 4333  df-int 4946
This theorem is referenced by:  intabs  5348  relintabex  43599  aiotavb  47107
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