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Theorem intnex 5341
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 5340 . . . 4 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
21necon1bbii 2979 . . 3 𝐴 ∈ V ↔ 𝐴 = ∅)
3 inteq 4953 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
4 int0 4966 . . . 4 ∅ = V
53, 4eqtrdi 2781 . . 3 (𝐴 = ∅ → 𝐴 = V)
62, 5sylbi 216 . 2 𝐴 ∈ V → 𝐴 = V)
7 vprc 5316 . . 3 ¬ V ∈ V
8 eleq1 2813 . . 3 ( 𝐴 = V → ( 𝐴 ∈ V ↔ V ∈ V))
97, 8mtbiri 326 . 2 ( 𝐴 = V → ¬ 𝐴 ∈ V)
106, 9impbii 208 1 𝐴 ∈ V ↔ 𝐴 = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wcel 2098  Vcvv 3461  c0 4322   cint 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4323  df-int 4951
This theorem is referenced by:  intabs  5345  relintabex  43153  aiotavb  46608
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