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Theorem iunn0 4026
 Description: There is a non-empty class in an indexed collection iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunn0
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem iunn0
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rexcom4 2878 . . 3
2 eliun 3973 . . . 4
32exbii 1582 . . 3
41, 3bitr4i 243 . 2
5 n0 3559 . . 3
65rexbii 2639 . 2
7 n0 3559 . 2
84, 6, 73bitr4i 268 1
 Colors of variables: wff setvar class Syntax hints:   wb 176  wex 1541   wcel 1710   wne 2516  wrex 2615  c0 3550  ciun 3969 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-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iun 3971 This theorem is referenced by: (None)
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