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Theorem iununir 3985
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 3833 . . . . . 6 (𝐵 = ∅ → 𝐵 = ∅)
2 uni0 3851 . . . . . 6 ∅ = ∅
31, 2eqtrdi 2238 . . . . 5 (𝐵 = ∅ → 𝐵 = ∅)
43uneq2d 3304 . . . 4 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∪ ∅))
5 un0 3471 . . . 4 (𝐴 ∪ ∅) = 𝐴
64, 5eqtrdi 2238 . . 3 (𝐵 = ∅ → (𝐴 𝐵) = 𝐴)
7 iuneq1 3914 . . . 4 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = 𝑥 ∈ ∅ (𝐴𝑥))
8 0iun 3959 . . . 4 𝑥 ∈ ∅ (𝐴𝑥) = ∅
97, 8eqtrdi 2238 . . 3 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = ∅)
106, 9eqeq12d 2204 . 2 (𝐵 = ∅ → ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) ↔ 𝐴 = ∅))
1110biimpcd 159 1 ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  cun 3142  c0 3437   cuni 3824   ciun 3901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-uni 3825  df-iun 3903
This theorem is referenced by: (None)
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