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Theorem 0iun 3705
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  U_  (/)  (/)

Proof of Theorem 0iun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rex0 3232 . . . 4  (/)
2 eliun 3652 . . . 4  U_  (/)  (/)
31, 2mtbir 595 . . 3  U_  (/)
4 noel 3222 . . 3  (/)
53, 42false 616 . 2  U_  (/)  (/)
65eqriv 2034 1  U_  (/)  (/)
Colors of variables: wff set class
Syntax hints:   wceq 1242   wcel 1390  wrex 2301   (/)c0 3218   U_ciun 3648
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-nul 3219  df-iun 3650
This theorem is referenced by:  iununir  3729  rdg0  5914
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