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Theorem uneqin 3188
 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 2997 . . . 4
2 unss 3117 . . . . 5
3 ssin 3159 . . . . . . 7
4 sstr 2953 . . . . . . 7
53, 4sylbir 125 . . . . . 6
6 ssin 3159 . . . . . . 7
7 simpl 102 . . . . . . 7
86, 7sylbir 125 . . . . . 6
95, 8anim12i 321 . . . . 5
102, 9sylbir 125 . . . 4
111, 10syl 14 . . 3
12 eqss 2960 . . 3
1311, 12sylibr 137 . 2
14 unidm 3086 . . . 4
15 inidm 3146 . . . 4
1614, 15eqtr4i 2063 . . 3
17 uneq2 3091 . . 3
18 ineq2 3132 . . 3
1916, 17, 183eqtr3a 2096 . 2
2013, 19impbii 117 1
 Colors of variables: wff set class Syntax hints:   wa 97   wb 98   wceq 1243   cun 2915   cin 2916   wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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