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Theorem unineq 3505
 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 2414 . . . . . . 7
2 elin 3219 . . . . . . 7
3 elin 3219 . . . . . . 7
41, 2, 33bitr3g 278 . . . . . 6
5 iba 489 . . . . . . 7
6 iba 489 . . . . . . 7
75, 6bibi12d 312 . . . . . 6
84, 7syl5ibr 212 . . . . 5
98adantld 453 . . . 4
10 uncom 3408 . . . . . . . . 9
11 uncom 3408 . . . . . . . . 9
1210, 11eqeq12i 2366 . . . . . . . 8
13 eleq2 2414 . . . . . . . 8
1412, 13sylbi 187 . . . . . . 7
15 elun 3220 . . . . . . 7
16 elun 3220 . . . . . . 7
1714, 15, 163bitr3g 278 . . . . . 6
18 biorf 394 . . . . . . 7
19 biorf 394 . . . . . . 7
2018, 19bibi12d 312 . . . . . 6
2117, 20syl5ibr 212 . . . . 5
2221adantrd 454 . . . 4
239, 22pm2.61i 156 . . 3
2423eqrdv 2351 . 2
25 uneq1 3411 . . 3
26 ineq1 3450 . . 3
2725, 26jca 518 . 2
2824, 27impbii 180 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wo 357   wa 358   wceq 1642   wcel 1710   cun 3207   cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214 This theorem is referenced by:  phiall  4618
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