Theorem iununir 4025

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 3873	. . . . . 6 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
2		uni0 3891	. . . . . 6 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2256	. . . . 5 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∅)
4	3	uneq2d 3335	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∅))
5		un0 3502	. . . 4 ⊢ (𝐴 ∪ ∅) = 𝐴
6	4, 5	eqtrdi 2256	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = 𝐴)
7		iuneq1 3954	. . . 4 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥))
8		0iun 3999	. . . 4 ⊢ ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥) = ∅
9	7, 8	eqtrdi 2256	. . 3 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∅)
10	6, 9	eqeq12d 2222	. 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 1373   ∪ cun 3172  ∅c0 3468  ∪ cuni 3864  ∪ ciun 3941

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649  df-uni 3865  df-iun 3943

This theorem is referenced by: (None)