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Theorem uniiunlem 3587
Description: A subset relationship useful for converting union to indexed union using dfiun2 4364 or dfiun2g 4362 and intersection to indexed intersection using dfiin2 4365. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Assertion
Ref Expression
uniiunlem
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem uniiunlem
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2461 . . . . . 6
21rexbidv 2968 . . . . 5
32cbvabv 2600 . . . 4
43sseq1i 3527 . . 3
5 r19.23v 2937 . . . . 5
65albii 1640 . . . 4
7 ralcom4 3128 . . . 4
8 abss 3568 . . . 4
96, 7, 83bitr4i 277 . . 3
104, 9bitr4i 252 . 2
11 nfv 1707 . . . . 5
12 eleq1 2529 . . . . 5
1311, 12ceqsalg 3134 . . . 4
1413ralimi 2850 . . 3
15 ralbi 2988 . . 3
1614, 15syl 16 . 2
1710, 16syl5rbb 258 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808  C_wss 3475
This theorem is referenced by:  mreiincl  14993  iunopn  19407  sigaclci  28132  dihglblem5  37025
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-3an 975  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
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