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Theorem iunsuc 6471
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 6392 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 iuneq1 5013 . . 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 5099 . 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 5096 . . 3 𝑥 ∈ {𝐴}𝐵 = 𝐶
87uneq2i 4175 . 2 ( 𝑥𝐴 𝐵 𝑥 ∈ {𝐴}𝐵) = ( 𝑥𝐴 𝐵𝐶)
93, 4, 83eqtri 2767 1 𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  {csn 4631   ciun 4996  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-iun 4998  df-suc 6392
This theorem is referenced by:  pwsdompw  10241
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