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Theorem intnex 5217
 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 5216 . . . 4 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
21necon1bbii 3055 . . 3 𝐴 ∈ V ↔ 𝐴 = ∅)
3 inteq 4855 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
4 int0 4866 . . . 4 ∅ = V
53, 4syl6eq 2871 . . 3 (𝐴 = ∅ → 𝐴 = V)
62, 5sylbi 219 . 2 𝐴 ∈ V → 𝐴 = V)
7 vprc 5195 . . 3 ¬ V ∈ V
8 eleq1 2898 . . 3 ( 𝐴 = V → ( 𝐴 ∈ V ↔ V ∈ V))
97, 8mtbiri 329 . 2 ( 𝐴 = V → ¬ 𝐴 ∈ V)
106, 9impbii 211 1 𝐴 ∈ V ↔ 𝐴 = V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1537   ∈ wcel 2114  Vcvv 3473  ∅c0 4269  ∩ cint 4852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rab 3134  df-v 3475  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4270  df-int 4853 This theorem is referenced by:  intabs  5221  relintabex  40076  aiotavb  43438
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