Theorem iununir 4049

Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
iununir	⊢ ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) → (𝐵 = ∅ → 𝐴 = ∅))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iununir

Step	Hyp	Ref	Expression
1		unieq 3897	. . . . . 6 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
2		uni0 3915	. . . . . 6 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2278	. . . . 5 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∅)
4	3	uneq2d 3358	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∅))
5		un0 3525	. . . 4 ⊢ (𝐴 ∪ ∅) = 𝐴
6	4, 5	eqtrdi 2278	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = 𝐴)
7		iuneq1 3978	. . . 4 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥))
8		0iun 4023	. . . 4 ⊢ ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥) = ∅
9	7, 8	eqtrdi 2278	. . 3 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∅)
10	6, 9	eqeq12d 2244	. 2 ⊢ (𝐵 = ∅ → ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) ↔ 𝐴 = ∅))
11	10	biimpcd 159	1 ⊢ ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) → (𝐵 = ∅ → 𝐴 = ∅))

Syntax hints:   → wi 4   = wceq 1395   ∪ cun 3195  ∅c0 3491  ∪ cuni 3888  ∪ ciun 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-uni 3889  df-iun 3967

This theorem is referenced by: (None)