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Theorem iununi 4415
 Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iununi
Distinct variable groups:   ,   ,

Proof of Theorem iununi
StepHypRef Expression
1 df-ne 2654 . . . . . . 7
2 iunconst 4339 . . . . . . 7
31, 2sylbir 213 . . . . . 6
4 iun0 4386 . . . . . . 7
5 id 22 . . . . . . . 8
65iuneq2d 4357 . . . . . . 7
74, 6, 53eqtr4a 2524 . . . . . 6
83, 7ja 161 . . . . 5
98eqcomd 2465 . . . 4
109uneq1d 3656 . . 3
11 uniiun 4383 . . . 4
1211uneq2i 3654 . . 3
13 iunun 4411 . . 3
1410, 12, 133eqtr4g 2523 . 2
15 unieq 4257 . . . . . . 7
16 uni0 4276 . . . . . . 7
1715, 16syl6eq 2514 . . . . . 6
1817uneq2d 3657 . . . . 5
19 un0 3810 . . . . 5
2018, 19syl6eq 2514 . . . 4
21 iuneq1 4344 . . . . 5
22 0iun 4387 . . . . 5
2321, 22syl6eq 2514 . . . 4
2420, 23eqeq12d 2479 . . 3
2524biimpcd 224 . 2
2614, 25impbii 188 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  =wceq 1395  =/=wne 2652  u.cun 3473   c0 3784  U.cuni 4249  U_ciun 4330 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-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-uni 4250  df-iun 4332
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