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Theorem iununi 3884
 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 2414 . . . . . . 7
2 iunconst 3811 . . . . . . 7
31, 2sylbir 206 . . . . . 6
4 iun0 3856 . . . . . . 7
5 id 21 . . . . . . . 8
65iuneq2d 3828 . . . . . . 7
74, 6, 53eqtr4a 2311 . . . . . 6
83, 7ja 155 . . . . 5
98eqcomd 2258 . . . 4
109uneq1d 3238 . . 3
11 uniiun 3853 . . . 4
1211uneq2i 3236 . . 3
13 iunun 3880 . . 3
1410, 12, 133eqtr4g 2310 . 2
15 unieq 3736 . . . . . . 7
16 uni0 3752 . . . . . . 7
1715, 16syl6eq 2301 . . . . . 6
1817uneq2d 3239 . . . . 5
19 un0 3386 . . . . 5
2018, 19syl6eq 2301 . . . 4
21 iuneq1 3816 . . . . 5
22 0iun 3857 . . . . 5
2321, 22syl6eq 2301 . . . 4
2420, 23eqeq12d 2267 . . 3
2524biimpcd 217 . 2
2614, 25impbii 182 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178   wceq 1619   wne 2412   cun 3076  c0 3362  cuni 3727  ciun 3803 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-sn 3550  df-uni 3728  df-iun 3805
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