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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

Proof of Theorem unissint
StepHypRef Expression
1 simpl 443 . . . . 5
2 df-ne 2518 . . . . . . 7
3 intssuni 3948 . . . . . . 7
42, 3sylbir 204 . . . . . 6
54adantl 452 . . . . 5
61, 5eqssd 3289 . . . 4
76ex 423 . . 3
87orrd 367 . 2
9 ssv 3291 . . . . 5
10 int0 3940 . . . . 5
119, 10sseqtr4i 3304 . . . 4
12 inteq 3929 . . . 4
1311, 12syl5sseqr 3320 . . 3
14 eqimss 3323 . . 3
1513, 14jaoi 368 . 2
168, 15impbii 180 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 176   wo 357   wa 358   wceq 1642   wne 2516  cvv 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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