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Theorem iununi 4167
 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 2600 . . . . . . 7
2 iunconst 4093 . . . . . . 7
31, 2sylbir 205 . . . . . 6
4 iun0 4139 . . . . . . 7
5 id 20 . . . . . . . 8
65iuneq2d 4110 . . . . . . 7
74, 6, 53eqtr4a 2493 . . . . . 6
83, 7ja 155 . . . . 5
98eqcomd 2440 . . . 4
109uneq1d 3492 . . 3
11 uniiun 4136 . . . 4
1211uneq2i 3490 . . 3
13 iunun 4163 . . 3
1410, 12, 133eqtr4g 2492 . 2
15 unieq 4016 . . . . . . 7
16 uni0 4034 . . . . . . 7
1715, 16syl6eq 2483 . . . . . 6
1817uneq2d 3493 . . . . 5
19 un0 3644 . . . . 5
2018, 19syl6eq 2483 . . . 4
21 iuneq1 4098 . . . . 5
22 0iun 4140 . . . . 5
2321, 22syl6eq 2483 . . . 4
2420, 23eqeq12d 2449 . . 3
2524biimpcd 216 . 2
2614, 25impbii 181 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wceq 1652   wne 2598   cun 3310  c0 3620  cuni 4007  ciun 4085 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-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-uni 4008  df-iun 4087
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