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Theorem iununi 5104
Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iununi ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iununi
StepHypRef Expression
1 df-ne 2939 . . . . . . 7 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2 iunconst 5006 . . . . . . 7 (𝐵 ≠ ∅ → 𝑥𝐵 𝐴 = 𝐴)
31, 2sylbir 235 . . . . . 6 𝐵 = ∅ → 𝑥𝐵 𝐴 = 𝐴)
4 iun0 5067 . . . . . . 7 𝑥𝐵 ∅ = ∅
5 id 22 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
65iuneq2d 5027 . . . . . . 7 (𝐴 = ∅ → 𝑥𝐵 𝐴 = 𝑥𝐵 ∅)
74, 6, 53eqtr4a 2801 . . . . . 6 (𝐴 = ∅ → 𝑥𝐵 𝐴 = 𝐴)
83, 7ja 186 . . . . 5 ((𝐵 = ∅ → 𝐴 = ∅) → 𝑥𝐵 𝐴 = 𝐴)
98eqcomd 2741 . . . 4 ((𝐵 = ∅ → 𝐴 = ∅) → 𝐴 = 𝑥𝐵 𝐴)
109uneq1d 4177 . . 3 ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 𝑥𝐵 𝑥) = ( 𝑥𝐵 𝐴 𝑥𝐵 𝑥))
11 uniiun 5063 . . . 4 𝐵 = 𝑥𝐵 𝑥
1211uneq2i 4175 . . 3 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
13 iunun 5098 . . 3 𝑥𝐵 (𝐴𝑥) = ( 𝑥𝐵 𝐴 𝑥𝐵 𝑥)
1410, 12, 133eqtr4g 2800 . 2 ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
15 unieq 4923 . . . . . . 7 (𝐵 = ∅ → 𝐵 = ∅)
16 uni0 4940 . . . . . . 7 ∅ = ∅
1715, 16eqtrdi 2791 . . . . . 6 (𝐵 = ∅ → 𝐵 = ∅)
1817uneq2d 4178 . . . . 5 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∪ ∅))
19 un0 4400 . . . . 5 (𝐴 ∪ ∅) = 𝐴
2018, 19eqtrdi 2791 . . . 4 (𝐵 = ∅ → (𝐴 𝐵) = 𝐴)
21 iuneq1 5013 . . . . 5 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = 𝑥 ∈ ∅ (𝐴𝑥))
22 0iun 5068 . . . . 5 𝑥 ∈ ∅ (𝐴𝑥) = ∅
2321, 22eqtrdi 2791 . . . 4 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = ∅)
2420, 23eqeq12d 2751 . . 3 (𝐵 = ∅ → ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) ↔ 𝐴 = ∅))
2524biimpcd 249 . 2 ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
2614, 25impbii 209 1 ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wne 2938  cun 3961  c0 4339   cuni 4912   ciun 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-uni 4913  df-iun 4998
This theorem is referenced by: (None)
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