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Theorem iununi 4642
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 2824 . . . . . . 7 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2 iunconst 4561 . . . . . . 7 (𝐵 ≠ ∅ → 𝑥𝐵 𝐴 = 𝐴)
31, 2sylbir 225 . . . . . 6 𝐵 = ∅ → 𝑥𝐵 𝐴 = 𝐴)
4 iun0 4608 . . . . . . 7 𝑥𝐵 ∅ = ∅
5 id 22 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
65iuneq2d 4579 . . . . . . 7 (𝐴 = ∅ → 𝑥𝐵 𝐴 = 𝑥𝐵 ∅)
74, 6, 53eqtr4a 2711 . . . . . 6 (𝐴 = ∅ → 𝑥𝐵 𝐴 = 𝐴)
83, 7ja 173 . . . . 5 ((𝐵 = ∅ → 𝐴 = ∅) → 𝑥𝐵 𝐴 = 𝐴)
98eqcomd 2657 . . . 4 ((𝐵 = ∅ → 𝐴 = ∅) → 𝐴 = 𝑥𝐵 𝐴)
109uneq1d 3799 . . 3 ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 𝑥𝐵 𝑥) = ( 𝑥𝐵 𝐴 𝑥𝐵 𝑥))
11 uniiun 4605 . . . 4 𝐵 = 𝑥𝐵 𝑥
1211uneq2i 3797 . . 3 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
13 iunun 4636 . . 3 𝑥𝐵 (𝐴𝑥) = ( 𝑥𝐵 𝐴 𝑥𝐵 𝑥)
1410, 12, 133eqtr4g 2710 . 2 ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
15 unieq 4476 . . . . . . 7 (𝐵 = ∅ → 𝐵 = ∅)
16 uni0 4497 . . . . . . 7 ∅ = ∅
1715, 16syl6eq 2701 . . . . . 6 (𝐵 = ∅ → 𝐵 = ∅)
1817uneq2d 3800 . . . . 5 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∪ ∅))
19 un0 4000 . . . . 5 (𝐴 ∪ ∅) = 𝐴
2018, 19syl6eq 2701 . . . 4 (𝐵 = ∅ → (𝐴 𝐵) = 𝐴)
21 iuneq1 4566 . . . . 5 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = 𝑥 ∈ ∅ (𝐴𝑥))
22 0iun 4609 . . . . 5 𝑥 ∈ ∅ (𝐴𝑥) = ∅
2321, 22syl6eq 2701 . . . 4 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = ∅)
2420, 23eqeq12d 2666 . . 3 (𝐵 = ∅ → ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) ↔ 𝐴 = ∅))
2524biimpcd 239 . 2 ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
2614, 25impbii 199 1 ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1523  wne 2823  cun 3605  c0 3948   cuni 4468   ciun 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469  df-iun 4554
This theorem is referenced by: (None)
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