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Theorem intnex 5351
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 5350 . . . 4 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
21necon1bbii 2988 . . 3 𝐴 ∈ V ↔ 𝐴 = ∅)
3 inteq 4954 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
4 int0 4967 . . . 4 ∅ = V
53, 4eqtrdi 2791 . . 3 (𝐴 = ∅ → 𝐴 = V)
62, 5sylbi 217 . 2 𝐴 ∈ V → 𝐴 = V)
7 vprc 5321 . . 3 ¬ V ∈ V
8 eleq1 2827 . . 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 1537  wcel 2106  Vcvv 3478  c0 4339   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-int 4952
This theorem is referenced by:  intabs  5355  relintabex  43571  aiotavb  47040
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