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Theorem unissint 3950
 Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 3963). (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 443 . . . . 5 ((A A ¬ A = ) → A A)
2 df-ne 2518 . . . . . . 7 (A ↔ ¬ A = )
3 intssuni 3948 . . . . . . 7 (AA A)
42, 3sylbir 204 . . . . . 6 A = A A)
54adantl 452 . . . . 5 ((A A ¬ A = ) → A A)
61, 5eqssd 3289 . . . 4 ((A A ¬ A = ) → A = A)
76ex 423 . . 3 (A A → (¬ A = A = A))
87orrd 367 . 2 (A A → (A = A = A))
9 ssv 3291 . . . . 5 A V
10 int0 3940 . . . . 5 = V
119, 10sseqtr4i 3304 . . . 4 A
12 inteq 3929 . . . 4 (A = A = )
1311, 12syl5sseqr 3320 . . 3 (A = A A)
14 eqimss 3323 . . 3 (A = AA A)
1513, 14jaoi 368 . 2 ((A = A = A) → A A)
168, 15impbii 180 1 (A A ↔ (A = A = A))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ≠ wne 2516  Vcvv 2859   ⊆ wss 3257  ∅c0 3550  ∪cuni 3891  ∩cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-uni 3892  df-int 3927 This theorem is referenced by: (None)
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