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Theorem unineq 3755
 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
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq2 2530 . . . . . . 7
2 elin 3683 . . . . . . 7
3 elin 3683 . . . . . . 7
41, 2, 33bitr3g 287 . . . . . 6
5 iba 503 . . . . . . 7
6 iba 503 . . . . . . 7
75, 6bibi12d 321 . . . . . 6
84, 7syl5ibr 221 . . . . 5
98adantld 467 . . . 4
10 uncom 3644 . . . . . . . . 9
11 uncom 3644 . . . . . . . . 9
1210, 11eqeq12i 2477 . . . . . . . 8
13 eleq2 2530 . . . . . . . 8
1412, 13sylbi 195 . . . . . . 7
15 elun 3641 . . . . . . 7
16 elun 3641 . . . . . . 7
1714, 15, 163bitr3g 287 . . . . . 6
18 biorf 405 . . . . . . 7
19 biorf 405 . . . . . . 7
2018, 19bibi12d 321 . . . . . 6
2117, 20syl5ibr 221 . . . . 5
2221adantrd 468 . . . 4
239, 22pm2.61i 164 . . 3
2423eqrdv 2454 . 2
25 uneq1 3647 . . 3
26 ineq1 3689 . . 3
2725, 26jca 532 . 2
2824, 27impbii 188 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wo 368   wa 369   wceq 1395   wcel 1819   cun 3469   cin 3470 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-in 3478 This theorem is referenced by: (None)
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