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Theorem iununi 5012
 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 3015 . . . . . . 7 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2 iunconst 4919 . . . . . . 7 (𝐵 ≠ ∅ → 𝑥𝐵 𝐴 = 𝐴)
31, 2sylbir 237 . . . . . 6 𝐵 = ∅ → 𝑥𝐵 𝐴 = 𝐴)
4 iun0 4976 . . . . . . 7 𝑥𝐵 ∅ = ∅
5 id 22 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
65iuneq2d 4939 . . . . . . 7 (𝐴 = ∅ → 𝑥𝐵 𝐴 = 𝑥𝐵 ∅)
74, 6, 53eqtr4a 2880 . . . . . 6 (𝐴 = ∅ → 𝑥𝐵 𝐴 = 𝐴)
83, 7ja 188 . . . . 5 ((𝐵 = ∅ → 𝐴 = ∅) → 𝑥𝐵 𝐴 = 𝐴)
98eqcomd 2825 . . . 4 ((𝐵 = ∅ → 𝐴 = ∅) → 𝐴 = 𝑥𝐵 𝐴)
109uneq1d 4136 . . 3 ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 𝑥𝐵 𝑥) = ( 𝑥𝐵 𝐴 𝑥𝐵 𝑥))
11 uniiun 4973 . . . 4 𝐵 = 𝑥𝐵 𝑥
1211uneq2i 4134 . . 3 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
13 iunun 5006 . . 3 𝑥𝐵 (𝐴𝑥) = ( 𝑥𝐵 𝐴 𝑥𝐵 𝑥)
1410, 12, 133eqtr4g 2879 . 2 ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
15 unieq 4838 . . . . . . 7 (𝐵 = ∅ → 𝐵 = ∅)
16 uni0 4857 . . . . . . 7 ∅ = ∅
1715, 16syl6eq 2870 . . . . . 6 (𝐵 = ∅ → 𝐵 = ∅)
1817uneq2d 4137 . . . . 5 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∪ ∅))
19 un0 4342 . . . . 5 (𝐴 ∪ ∅) = 𝐴
2018, 19syl6eq 2870 . . . 4 (𝐵 = ∅ → (𝐴 𝐵) = 𝐴)
21 iuneq1 4926 . . . . 5 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = 𝑥 ∈ ∅ (𝐴𝑥))
22 0iun 4977 . . . . 5 𝑥 ∈ ∅ (𝐴𝑥) = ∅
2321, 22syl6eq 2870 . . . 4 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = ∅)
2420, 23eqeq12d 2835 . . 3 (𝐵 = ∅ → ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) ↔ 𝐴 = ∅))
2524biimpcd 251 . 2 ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
2614, 25impbii 211 1 ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1530   ≠ wne 3014   ∪ cun 3932  ∅c0 4289  ∪ cuni 4830  ∪ ciun 4910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-uni 4831  df-iun 4912 This theorem is referenced by: (None)
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