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Theorem 0iun 3688
 Description: An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
0iun x A = ∅

Proof of Theorem 0iun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 rex0 3215 . . . 4 ¬ x y A
2 eliun 3635 . . . 4 (y x Ax y A)
31, 2mtbir 583 . . 3 ¬ y x A
4 noel 3205 . . 3 ¬ y
53, 42false 604 . 2 (y x Ay ∅)
65eqriv 2019 1 x A = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  ∅c0 3201  ∪ ciun 3631 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-nul 3202  df-iun 3633 This theorem is referenced by:  iununir  3712  rdgi0g  5886  rdg0  5895
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