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Theorem unissint 4609
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4622). (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 2897 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3 intssuni 4607 . . . . . . 7 (𝐴 ≠ ∅ → 𝐴 𝐴)
42, 3sylbir 225 . . . . . 6 𝐴 = ∅ → 𝐴 𝐴)
54adantl 473 . . . . 5 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 𝐴)
61, 5eqssd 3726 . . . 4 (( 𝐴 𝐴 ∧ ¬ 𝐴 = ∅) → 𝐴 = 𝐴)
76ex 449 . . 3 ( 𝐴 𝐴 → (¬ 𝐴 = ∅ → 𝐴 = 𝐴))
87orrd 392 . 2 ( 𝐴 𝐴 → (𝐴 = ∅ ∨ 𝐴 = 𝐴))
9 ssv 3731 . . . . 5 𝐴 ⊆ V
10 int0 4598 . . . . 5 ∅ = V
119, 10sseqtr4i 3744 . . . 4 𝐴
12 inteq 4586 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1311, 12syl5sseqr 3760 . . 3 (𝐴 = ∅ → 𝐴 𝐴)
14 eqimss 3763 . . 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 1596  wne 2896  Vcvv 3304  wss 3680  c0 4023   cuni 4544   cint 4583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694  df-nul 4024  df-uni 4545  df-int 4584
This theorem is referenced by: (None)
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