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Theorem unineq 3583
 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 2496 . . . . . . 7
2 elin 3522 . . . . . . 7
3 elin 3522 . . . . . . 7
41, 2, 33bitr3g 279 . . . . . 6
5 iba 490 . . . . . . 7
6 iba 490 . . . . . . 7
75, 6bibi12d 313 . . . . . 6
84, 7syl5ibr 213 . . . . 5
98adantld 454 . . . 4
10 uncom 3483 . . . . . . . . 9
11 uncom 3483 . . . . . . . . 9
1210, 11eqeq12i 2448 . . . . . . . 8
13 eleq2 2496 . . . . . . . 8
1412, 13sylbi 188 . . . . . . 7
15 elun 3480 . . . . . . 7
16 elun 3480 . . . . . . 7
1714, 15, 163bitr3g 279 . . . . . 6
18 biorf 395 . . . . . . 7
19 biorf 395 . . . . . . 7
2018, 19bibi12d 313 . . . . . 6
2117, 20syl5ibr 213 . . . . 5
2221adantrd 455 . . . 4
239, 22pm2.61i 158 . . 3
2423eqrdv 2433 . 2
25 uneq1 3486 . . 3
26 ineq1 3527 . . 3
2725, 26jca 519 . 2
2824, 27impbii 181 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359   wceq 1652   wcel 1725   cun 3310   cin 3311 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319
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