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Theorem unineq 3326
 Description: Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)
Assertion
Ref Expression
unineq

Proof of Theorem unineq
StepHypRef Expression
1 eleq2 2314 . . . . . . 7
2 elin 3266 . . . . . . 7
3 elin 3266 . . . . . . 7
41, 2, 33bitr3g 280 . . . . . 6
5 iba 491 . . . . . . 7
6 iba 491 . . . . . . 7
75, 6bibi12d 314 . . . . . 6
84, 7syl5ibr 214 . . . . 5
98adantld 455 . . . 4
10 uncom 3229 . . . . . . . . 9
11 uncom 3229 . . . . . . . . 9
1210, 11eqeq12i 2266 . . . . . . . 8
13 eleq2 2314 . . . . . . . 8
1412, 13sylbi 189 . . . . . . 7
15 elun 3226 . . . . . . 7
16 elun 3226 . . . . . . 7
1714, 15, 163bitr3g 280 . . . . . 6
18 biorf 396 . . . . . . 7
19 biorf 396 . . . . . . 7
2018, 19bibi12d 314 . . . . . 6
2117, 20syl5ibr 214 . . . . 5
2221adantrd 456 . . . 4
239, 22pm2.61i 158 . . 3
2423eqrdv 2251 . 2
25 uneq1 3232 . . 3
26 ineq1 3271 . . 3
2725, 26jca 520 . 2
2824, 27impbii 182 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178   wo 359   wa 360   wceq 1619   wcel 1621   cun 3076   cin 3077 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-in 3085
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