Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  unissint Structured version   Visualization version   GIF version

Theorem unissint 4865
 Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4878). (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 486 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
2 df-ne 2991 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3 intssuni 4863 . . . . . . 7 (𝐴 ≠ ∅ → 𝐴 𝐴)
42, 3sylbir 238 . . . . . 6 𝐴 = ∅ → 𝐴 𝐴)
54adantl 485 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
61, 5eqssd 3935 . . . 4 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 = 𝐴)
76ex 416 . . 3 ( 𝐴 𝐴 → (¬ 𝐴 = ∅ → 𝐴 = 𝐴))
87orrd 860 . 2 ( 𝐴 𝐴 → (𝐴 = ∅ ∨ 𝐴 = 𝐴))
9 ssv 3942 . . . . 5 𝐴 ⊆ V
10 int0 4855 . . . . 5 ∅ = V
119, 10sseqtrri 3955 . . . 4 𝐴
12 inteq 4844 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1311, 12sseqtrrid 3971 . . 3 (𝐴 = ∅ → 𝐴 𝐴)
14 eqimss 3974 . . 3 ( 𝐴 = 𝐴 𝐴 𝐴)
1513, 14jaoi 854 . 2 ((𝐴 = ∅ ∨ 𝐴 = 𝐴) → 𝐴 𝐴)
168, 15impbii 212 1 ( 𝐴 𝐴 ↔ (𝐴 = ∅ ∨ 𝐴 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ≠ wne 2990  Vcvv 3444   ⊆ wss 3884  ∅c0 4246  ∪ cuni 4803  ∩ cint 4841 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247  df-uni 4804  df-int 4842 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator