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Theorem iunsuc 4157
 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 = ( 𝑥𝐴 𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunsuc
StepHypRef Expression
1 df-suc 4108 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 iuneq1 3670 . . 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 3735 . 2 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵 = ( 𝑥𝐴 𝐵 𝑥 ∈ {𝐴}𝐵)
5 iunsuc.1 . . . 4 𝐴 ∈ V
6 iunsuc.2 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
75, 6iunxsn 3733 . . 3 𝑥 ∈ {𝐴}𝐵 = 𝐶
87uneq2i 3094 . 2 ( 𝑥𝐴 𝐵 𝑥 ∈ {𝐴}𝐵) = ( 𝑥𝐴 𝐵𝐶)
93, 4, 83eqtri 2064 1 𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915  {csn 3375  ∪ ciun 3657  suc csuc 4102 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 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-iun 3659  df-suc 4108 This theorem is referenced by: (None)
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