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Theorem iunpw 6614
 Description: An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
Hypothesis
Ref Expression
iunpw.1
Assertion
Ref Expression
iunpw
Distinct variable group:   ,

Proof of Theorem iunpw
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sseq2 3525 . . . . . . . 8
21biimprcd 225 . . . . . . 7
32reximdv 2931 . . . . . 6
43com12 31 . . . . 5
5 ssiun 4372 . . . . . 6
6 uniiun 4383 . . . . . 6
75, 6syl6sseqr 3550 . . . . 5
84, 7impbid1 203 . . . 4
9 selpw 4019 . . . 4
10 eliun 4335 . . . . 5
11 selpw 4019 . . . . . 6
1211rexbii 2959 . . . . 5
1310, 12bitri 249 . . . 4
148, 9, 133bitr4g 288 . . 3
1514eqrdv 2454 . 2
16 ssid 3522 . . . . 5
17 iunpw.1 . . . . . . . 8
1817uniex 6596 . . . . . . 7
1918elpw 4018 . . . . . 6
20 eleq2 2530 . . . . . 6
2119, 20syl5bbr 259 . . . . 5
2216, 21mpbii 211 . . . 4
23 eliun 4335 . . . 4
2422, 23sylib 196 . . 3
25 elssuni 4279 . . . . . . 7
26 elpwi 4021 . . . . . . 7
2725, 26anim12i 566 . . . . . 6
28 eqss 3518 . . . . . 6
2927, 28sylibr 212 . . . . 5
3029ex 434 . . . 4
3130reximia 2923 . . 3
3224, 31syl 16 . 2
3315, 32impbii 188 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  E.wrex 2808   cvv 3109  C_wss 3475  ~Pcpw 4012  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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-un 6592 This theorem depends on definitions:  df-bi 185  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-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-pw 4014  df-uni 4250  df-iun 4332
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