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Theorem uniiunlem 3353
 Description: A subset relationship useful for converting union to indexed union using dfiun2 4001 or dfiun2g 3999 and intersection to indexed intersection using dfiin2 4002. (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 2359 . . . . . 6
21rexbidv 2635 . . . . 5
32cbvabv 2472 . . . 4
43sseq1i 3295 . . 3
5 r19.23v 2730 . . . . 5
65albii 1566 . . . 4
7 ralcom4 2877 . . . 4
8 abss 3335 . . . 4
96, 7, 83bitr4i 268 . . 3
104, 9bitr4i 243 . 2
11 nfv 1619 . . . . 5
12 eleq1 2413 . . . . 5
1311, 12ceqsalg 2883 . . . 4
1413ralimi 2689 . . 3
15 ralbi 2750 . . 3
1614, 15syl 15 . 2
1710, 16syl5rbb 249 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176  wal 1540   wceq 1642   wcel 1710  cab 2339  wral 2614  wrex 2615   wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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