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Theorem iunsuc 6045
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 5969 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 iuneq1 4754 . . 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 4826 . 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 4823 . . 3 𝑥 ∈ {𝐴}𝐵 = 𝐶
87uneq2i 3991 . 2 ( 𝑥𝐴 𝐵 𝑥 ∈ {𝐴}𝐵) = ( 𝑥𝐴 𝐵𝐶)
93, 4, 83eqtri 2853 1 𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  Vcvv 3414  cun 3796  {csn 4397   ciun 4740  suc csuc 5965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-sbc 3663  df-un 3803  df-in 3805  df-ss 3812  df-sn 4398  df-iun 4742  df-suc 5969
This theorem is referenced by:  pwsdompw  9341
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