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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 A V
iunsuc.2 (x = AB = 𝐶)
Assertion
Ref Expression
iunsuc x suc AB = ( x A B𝐶)
Distinct variable groups:   x,A   x,𝐶
Allowed substitution hint:   B(x)

Proof of Theorem iunsuc
StepHypRef Expression
1 df-suc 4074 . . 3 suc A = (A ∪ {A})
2 iuneq1 3661 . . 3 (suc A = (A ∪ {A}) → x suc AB = x (A ∪ {A})B)
31, 2ax-mp 7 . 2 x suc AB = x (A ∪ {A})B
4 iunxun 3726 . 2 x (A ∪ {A})B = ( x A B x {A}B)
5 iunsuc.1 . . . 4 A V
6 iunsuc.2 . . . 4 (x = AB = 𝐶)
75, 6iunxsn 3724 . . 3 x {A}B = 𝐶
87uneq2i 3088 . 2 ( x A B x {A}B) = ( x A B𝐶)
93, 4, 83eqtri 2061 1 x suc AB = ( x A B𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909  {csn 3367   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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