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Theorem iunsuc 4123
Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
iunsuc.1  _V
iunsuc.2  C
Assertion
Ref Expression
iunsuc  U_  suc  U_  u.  C
Distinct variable groups:   ,   , C
Allowed substitution hint:   ()

Proof of Theorem iunsuc
StepHypRef Expression
1 df-suc 4074 . . 3  suc  u.  { }
2 iuneq1 3661 . . 3  suc  u.  { }  U_  suc  U_  u.  { }
31, 2ax-mp 7 . 2  U_  suc  U_  u.  { }
4 iunxun 3726 . 2  U_  u.  { }  U_  u.  U_ 
{ }
5 iunsuc.1 . . . 4  _V
6 iunsuc.2 . . . 4  C
75, 6iunxsn 3724 . . 3  U_  { }  C
87uneq2i 3088 . 2  U_  u.  U_  { }  U_  u.  C
93, 4, 83eqtri 2061 1  U_  suc  U_  u.  C
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390   _Vcvv 2551    u. cun 2909   {csn 3367   U_ciun 3648   suc csuc 4068
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-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-bndl 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-3an 886  df-tru 1245  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-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-iun 3650  df-suc 4074
This theorem is referenced by: (None)
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