Theorem iunsuc 6419

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 6338	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		iuneq1 4972	. . 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 5058	. 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 5055	. . 3 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶
8	7	uneq2i 4128	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)
9	3, 4, 8	3eqtri 2756	1 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912  {csn 4589  ∪ ciun 4955  suc csuc 6334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-v 3449  df-un 3919  df-ss 3931  df-sn 4590  df-iun 4957  df-suc 6338

This theorem is referenced by:  pwsdompw  10156