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Theorem iunsuc 4221
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 4172 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 iuneq1 3726 . . 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 3791 . 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 3789 . . 3 𝑥 ∈ {𝐴}𝐵 = 𝐶
87uneq2i 3140 . 2 ( 𝑥𝐴 𝐵 𝑥 ∈ {𝐴}𝐵) = ( 𝑥𝐴 𝐵𝐶)
93, 4, 83eqtri 2109 1 𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  Vcvv 2615  cun 2986  {csn 3431   ciun 3713  suc csuc 4166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-iun 3715  df-suc 4172
This theorem is referenced by: (None)
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