Theorem iunsuc 6452

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 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iunsuc	⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunsuc

Step	Hyp	Ref	Expression
1		df-suc 6370	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		iuneq1 4975	. . 3 ⊢ (suc 𝐴 = (𝐴 ∪ {𝐴}) → ∪ 𝑥 ∈ suc 𝐴𝐵 = ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵)
3	1, 2	ax-mp 5	. 2 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵
4		iunxun 5062	. 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵)
5		iunsuc.1	. . . 4 ⊢ 𝐴 ∈ V
6		iunsuc.2	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
7	5, 6	iunxsn 5059	. . 3 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶
8	7	uneq2i 4119	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)
9	3, 4, 8	3eqtri 2792	1 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∪ cun 3904  {csn 4591  ∪ ciun 4958  suc csuc 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-v 3459  df-un 3911  df-ss 3923  df-sn 4592  df-iun 4960  df-suc 6370

This theorem is used by:  pwsdompw  10202