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Theorem ssiun 4133
 Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem ssiun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3342 . . . . 5
21reximi 2813 . . . 4
3 r19.37av 2858 . . . 4
42, 3syl 16 . . 3
5 eliun 4097 . . 3
64, 5syl6ibr 219 . 2
76ssrdv 3354 1
 Colors of variables: wff set class Syntax hints:   wi 4   wcel 1725  wrex 2706   wss 3320  ciun 4093 This theorem is referenced by:  iunss2  4136  iunpwss  4180  iunpw  4759  onfununi  6603  oen0  6829  trcl  7664  rtrclreclem.refl  25144  rtrclreclem.subset  25145  trpredtr  25508  dftrpred3g  25511  wfrlem9  25546  frrlem5e  25590 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 2417 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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-iun 4095
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