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Theorem unissint 4931
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4946). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
unissint ( 𝐴 𝐴 ↔ (𝐴 = ∅ ∨ 𝐴 = 𝐴))

Proof of Theorem unissint
StepHypRef Expression
1 simpl 483 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
2 df-ne 2942 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3 intssuni 4929 . . . . . . 7 (𝐴 ≠ ∅ → 𝐴 𝐴)
42, 3sylbir 234 . . . . . 6 𝐴 = ∅ → 𝐴 𝐴)
54adantl 482 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
61, 5eqssd 3959 . . . 4 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 = 𝐴)
76ex 413 . . 3 ( 𝐴 𝐴 → (¬ 𝐴 = ∅ → 𝐴 = 𝐴))
87orrd 861 . 2 ( 𝐴 𝐴 → (𝐴 = ∅ ∨ 𝐴 = 𝐴))
9 ssv 3966 . . . . 5 𝐴 ⊆ V
10 int0 4921 . . . . 5 ∅ = V
119, 10sseqtrri 3979 . . . 4 𝐴
12 inteq 4908 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1311, 12sseqtrrid 3995 . . 3 (𝐴 = ∅ → 𝐴 𝐴)
14 eqimss 3998 . . 3 ( 𝐴 = 𝐴 𝐴 𝐴)
1513, 14jaoi 855 . 2 ((𝐴 = ∅ ∨ 𝐴 = 𝐴) → 𝐴 𝐴)
168, 15impbii 208 1 ( 𝐴 𝐴 ↔ (𝐴 = ∅ ∨ 𝐴 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  wo 845   = wceq 1541  wne 2941  Vcvv 3443  wss 3908  c0 4280   cuni 4863   cint 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3445  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4281  df-uni 4864  df-int 4906
This theorem is referenced by: (None)
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