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Theorem iununir 3896
 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) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iununir
StepHypRef Expression
1 unieq 3745 . . . . . 6 (𝐵 = ∅ → 𝐵 = ∅)
2 uni0 3763 . . . . . 6 ∅ = ∅
31, 2syl6eq 2188 . . . . 5 (𝐵 = ∅ → 𝐵 = ∅)
43uneq2d 3230 . . . 4 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∪ ∅))
5 un0 3396 . . . 4 (𝐴 ∪ ∅) = 𝐴
64, 5syl6eq 2188 . . 3 (𝐵 = ∅ → (𝐴 𝐵) = 𝐴)
7 iuneq1 3826 . . . 4 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = 𝑥 ∈ ∅ (𝐴𝑥))
8 0iun 3870 . . . 4 𝑥 ∈ ∅ (𝐴𝑥) = ∅
97, 8syl6eq 2188 . . 3 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = ∅)
106, 9eqeq12d 2154 . 2 (𝐵 = ∅ → ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) ↔ 𝐴 = ∅))
1110biimpcd 158 1 ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∪ cun 3069  ∅c0 3363  ∪ cuni 3736  ∪ ciun 3813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-uni 3737  df-iun 3815 This theorem is referenced by: (None)
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