Theorem iunsuc 4380

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 4331	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		iuneq1 3862	. . 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 3928	. 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 3925	. . 3 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶
8	7	uneq2i 3258	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)
9	3, 4, 8	3eqtri 2182	1 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)

Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128  Vcvv 2712   ∪ cun 3100  {csn 3560  ∪ ciun 3849  suc csuc 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-iun 3851  df-suc 4331

This theorem is referenced by: (None)