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Theorem iunsuc 6348
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 6272 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 iuneq1 4940 . . 3 (suc 𝐴 = (𝐴 ∪ {𝐴}) → 𝑥 ∈ suc 𝐴𝐵 = 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵)
31, 2ax-mp 5 . 2 𝑥 ∈ suc 𝐴𝐵 = 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵
4 iunxun 5023 . 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 5020 . . 3 𝑥 ∈ {𝐴}𝐵 = 𝐶
87uneq2i 4094 . 2 ( 𝑥𝐴 𝐵 𝑥 ∈ {𝐴}𝐵) = ( 𝑥𝐴 𝐵𝐶)
93, 4, 83eqtri 2770 1 𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  Vcvv 3432  cun 3885  {csn 4561   ciun 4924  suc csuc 6268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-iun 4926  df-suc 6272
This theorem is referenced by:  pwsdompw  9960
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