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Theorem iunsuc 4126
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 4077 . . 3 suc A = (A ∪ {A})
2 iuneq1 3664 . . 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 3729 . 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 3727 . . 3 x {A}B = 𝐶
87uneq2i 3091 . 2 ( x A B x {A}B) = ( x A B𝐶)
93, 4, 83eqtri 2064 1 x suc AB = ( x A B𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243   wcel 1393  Vcvv 2554  cun 2912  {csn 3370   ciun 3651  suc csuc 4071
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-sn 3376  df-iun 3653  df-suc 4077
This theorem is referenced by: (None)
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