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Theorem uneqin 3327
 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 3151 . . . 4
2 unss 3250 . . . . 5
3 ssin 3298 . . . . . . 7
4 sstr 3105 . . . . . . 7
53, 4sylbir 134 . . . . . 6
6 ssin 3298 . . . . . . 7
7 simpl 108 . . . . . . 7
86, 7sylbir 134 . . . . . 6
95, 8anim12i 336 . . . . 5
102, 9sylbir 134 . . . 4
111, 10syl 14 . . 3
12 eqss 3112 . . 3
1311, 12sylibr 133 . 2
14 unidm 3219 . . . 4
15 inidm 3285 . . . 4
1614, 15eqtr4i 2163 . . 3
17 uneq2 3224 . . 3
18 ineq2 3271 . . 3
1916, 17, 183eqtr3a 2196 . 2
2013, 19impbii 125 1
 Colors of variables: wff set class Syntax hints:   wa 103   wb 104   wceq 1331   cun 3069   cin 3070   wss 3071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084 This theorem is referenced by: (None)
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