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Theorem uneqin 3748
 Description: Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
uneqin

Proof of Theorem uneqin
StepHypRef Expression
1 eqimss 3555 . . . 4
2 unss 3677 . . . . 5
3 ssin 3719 . . . . . . 7
4 sstr 3511 . . . . . . 7
53, 4sylbir 213 . . . . . 6
6 ssin 3719 . . . . . . 7
7 simpl 457 . . . . . . 7
86, 7sylbir 213 . . . . . 6
95, 8anim12i 566 . . . . 5
102, 9sylbir 213 . . . 4
111, 10syl 16 . . 3
12 eqss 3518 . . 3
1311, 12sylibr 212 . 2
14 unidm 3646 . . . 4
15 inidm 3706 . . . 4
1614, 15eqtr4i 2489 . . 3
17 uneq2 3651 . . 3
18 ineq2 3693 . . 3
1916, 17, 183eqtr3a 2522 . 2
2013, 19impbii 188 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  u.cun 3473  i^icin 3474  C_wss 3475 This theorem is referenced by:  uniintsn  4324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482  df-ss 3489
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