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Theorem uniiunlem 3423
 Description: A subset relationship useful for converting union to indexed union using dfiun2 4117 or dfiun2g 4115 and intersection to indexed intersection using dfiin2 4118. (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 2441 . . . . . 6
21rexbidv 2718 . . . . 5
32cbvabv 2554 . . . 4
43sseq1i 3364 . . 3
5 r19.23v 2814 . . . . 5
65albii 1575 . . . 4
7 ralcom4 2966 . . . 4
8 abss 3404 . . . 4
96, 7, 83bitr4i 269 . . 3
104, 9bitr4i 244 . 2
11 nfv 1629 . . . . 5
12 eleq1 2495 . . . . 5
1311, 12ceqsalg 2972 . . . 4
1413ralimi 2773 . . 3
15 ralbi 2834 . . 3
1614, 15syl 16 . 2
1710, 16syl5rbb 250 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177  wal 1549   wceq 1652   wcel 1725  cab 2421  wral 2697  wrex 2698   wss 3312 This theorem is referenced by:  mreiincl  13813  iunopn  16963  sigaclci  24507  dihglblem5  32033 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-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326
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