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Theorem unineq 3747
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 3686 . . . . . . 7
3 elin 3686 . . . . . . 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 3647 . . . . . . . . 9
11 uncom 3647 . . . . . . . . 9
1210, 11eqeq12i 2477 . . . . . . . 8
13 eleq2 2530 . . . . . . . 8
1412, 13sylbi 195 . . . . . . 7
15 elun 3644 . . . . . . 7
16 elun 3644 . . . . . . 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 3650 . . 3
26 ineq1 3692 . . 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  e.wcel 1818  u.cun 3473  i^icin 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482
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