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

Theorem unissint 4645
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4658). (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 474 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
2 df-ne 2925 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3 intssuni 4643 . . . . . . 7 (𝐴 ≠ ∅ → 𝐴 𝐴)
42, 3sylbir 225 . . . . . 6 𝐴 = ∅ → 𝐴 𝐴)
54adantl 473 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
61, 5eqssd 3753 . . . 4 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 = 𝐴)
76ex 449 . . 3 ( 𝐴 𝐴 → (¬ 𝐴 = ∅ → 𝐴 = 𝐴))
87orrd 392 . 2 ( 𝐴 𝐴 → (𝐴 = ∅ ∨ 𝐴 = 𝐴))
9 ssv 3758 . . . . 5 𝐴 ⊆ V
10 int0 4634 . . . . 5 ∅ = V
119, 10sseqtr4i 3771 . . . 4 𝐴
12 inteq 4622 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1311, 12syl5sseqr 3787 . . 3 (𝐴 = ∅ → 𝐴 𝐴)
14 eqimss 3790 . . 3 ( 𝐴 = 𝐴 𝐴 𝐴)
1513, 14jaoi 393 . 2 ((𝐴 = ∅ ∨ 𝐴 = 𝐴) → 𝐴 𝐴)
168, 15impbii 199 1 ( 𝐴 𝐴 ↔ (𝐴 = ∅ ∨ 𝐴 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383   = wceq 1624  wne 2924  Vcvv 3332  wss 3707  c0 4050   cuni 4580   cint 4619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-v 3334  df-dif 3710  df-in 3714  df-ss 3721  df-nul 4051  df-uni 4581  df-int 4620
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator