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Theorem iun0 4022
 Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iun0 x A =

Proof of Theorem iun0
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 noel 3554 . . . . . 6 ¬ y
21a1i 10 . . . . 5 (x A → ¬ y )
32nrex 2716 . . . 4 ¬ x A y
4 eliun 3973 . . . 4 (y x A x A y )
53, 4mtbir 290 . . 3 ¬ y x A
65, 12false 339 . 2 (y x A y )
76eqriv 2350 1 x A =
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∅c0 3550  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-dif 3215  df-nul 3551  df-iun 3971 This theorem is referenced by:  iununi  4050
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