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Theorem uniiunlem 3526
 Description: A subset relationship useful for converting union to indexed union using dfiun2 4304 or dfiun2g 4302 and intersection to indexed intersection using dfiin2 4305. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Assertion
Ref Expression
uniiunlem
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   (,)

Proof of Theorem uniiunlem
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2406 . . . . . 6
21rexbidv 2917 . . . . 5
32cbvabv 2545 . . . 4
43sseq1i 3465 . . 3
5 r19.23v 2883 . . . . 5
65albii 1661 . . . 4
7 ralcom4 3077 . . . 4
8 abss 3507 . . . 4
96, 7, 83bitr4i 277 . . 3
104, 9bitr4i 252 . 2
11 nfv 1728 . . . . 5
12 eleq1 2474 . . . . 5
1311, 12ceqsalg 3083 . . . 4
1413ralimi 2796 . . 3
15 ralbi 2937 . . 3
1614, 15syl 17 . 2
1710, 16syl5rbb 258 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184  wal 1403   wceq 1405   wcel 1842  cab 2387  wral 2753  wrex 2754   wss 3413 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-v 3060  df-in 3420  df-ss 3427 This theorem is referenced by:  mreiincl  15102  iunopn  19591  sigaclci  28460  dihglblem5  34299
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