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Theorem intnex 5011
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 5010 . . . 4 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
21necon1bbii 3018 . . 3 𝐴 ∈ V ↔ 𝐴 = ∅)
3 inteq 4668 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
4 int0 4679 . . . 4 ∅ = V
53, 4syl6eq 2847 . . 3 (𝐴 = ∅ → 𝐴 = V)
62, 5sylbi 209 . 2 𝐴 ∈ V → 𝐴 = V)
7 vprc 4990 . . 3 ¬ V ∈ V
8 eleq1 2864 . . 3 ( 𝐴 = V → ( 𝐴 ∈ V ↔ V ∈ V))
97, 8mtbiri 319 . 2 ( 𝐴 = V → ¬ 𝐴 ∈ V)
106, 9impbii 201 1 𝐴 ∈ V ↔ 𝐴 = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198   = wceq 1653  wcel 2157  Vcvv 3383  c0 4113   cint 4665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-v 3385  df-dif 3770  df-in 3774  df-ss 3781  df-nul 4114  df-int 4666
This theorem is referenced by:  intabs  5015  relintabex  38658  aiotavb  41927
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