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Theorem unissint 4971
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4984). (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 482 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
2 df-ne 2940 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3 intssuni 4969 . . . . . . 7 (𝐴 ≠ ∅ → 𝐴 𝐴)
42, 3sylbir 235 . . . . . 6 𝐴 = ∅ → 𝐴 𝐴)
54adantl 481 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
61, 5eqssd 4000 . . . 4 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 = 𝐴)
76ex 412 . . 3 ( 𝐴 𝐴 → (¬ 𝐴 = ∅ → 𝐴 = 𝐴))
87orrd 863 . 2 ( 𝐴 𝐴 → (𝐴 = ∅ ∨ 𝐴 = 𝐴))
9 ssv 4007 . . . . 5 𝐴 ⊆ V
10 int0 4961 . . . . 5 ∅ = V
119, 10sseqtrri 4032 . . . 4 𝐴
12 inteq 4948 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1311, 12sseqtrrid 4026 . . 3 (𝐴 = ∅ → 𝐴 𝐴)
14 eqimss 4041 . . 3 ( 𝐴 = 𝐴 𝐴 𝐴)
1513, 14jaoi 857 . 2 ((𝐴 = ∅ ∨ 𝐴 = 𝐴) → 𝐴 𝐴)
168, 15impbii 209 1 ( 𝐴 𝐴 ↔ (𝐴 = ∅ ∨ 𝐴 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847   = wceq 1539  wne 2939  Vcvv 3479  wss 3950  c0 4332   cuni 4906   cint 4945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3953  df-ss 3967  df-nul 4333  df-uni 4907  df-int 4946
This theorem is referenced by: (None)
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