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Theorem unissint 4074
 Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4087). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
unissint

Proof of Theorem unissint
StepHypRef Expression
1 simpl 444 . . . . 5
2 df-ne 2601 . . . . . . 7
3 intssuni 4072 . . . . . . 7
42, 3sylbir 205 . . . . . 6
54adantl 453 . . . . 5
61, 5eqssd 3365 . . . 4
76ex 424 . . 3
87orrd 368 . 2
9 ssv 3368 . . . . 5
10 int0 4064 . . . . 5
119, 10sseqtr4i 3381 . . . 4
12 inteq 4053 . . . 4
1311, 12syl5sseqr 3397 . . 3
14 eqimss 3400 . . 3
1513, 14jaoi 369 . 2
168, 15impbii 181 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177   wo 358   wa 359   wceq 1652   wne 2599  cvv 2956   wss 3320  c0 3628  cuni 4015  cint 4050 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-uni 4016  df-int 4051
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