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Theorem iununir 3780
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 3631 . . . . . 6 (𝐵 = ∅ → 𝐵 = ∅)
2 uni0 3649 . . . . . 6 ∅ = ∅
31, 2syl6eq 2131 . . . . 5 (𝐵 = ∅ → 𝐵 = ∅)
43uneq2d 3137 . . . 4 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∪ ∅))
5 un0 3295 . . . 4 (𝐴 ∪ ∅) = 𝐴
64, 5syl6eq 2131 . . 3 (𝐵 = ∅ → (𝐴 𝐵) = 𝐴)
7 iuneq1 3712 . . . 4 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = 𝑥 ∈ ∅ (𝐴𝑥))
8 0iun 3756 . . . 4 𝑥 ∈ ∅ (𝐴𝑥) = ∅
97, 8syl6eq 2131 . . 3 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = ∅)
106, 9eqeq12d 2097 . 2 (𝐵 = ∅ → ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) ↔ 𝐴 = ∅))
1110biimpcd 157 1 ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  cun 2981  c0 3268   cuni 3622   ciun 3699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-sn 3423  df-uni 3623  df-iun 3701
This theorem is referenced by: (None)
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