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Theorem iununir 3834
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 3684 . . . . . 6 (𝐵 = ∅ → 𝐵 = ∅)
2 uni0 3702 . . . . . 6 ∅ = ∅
31, 2syl6eq 2143 . . . . 5 (𝐵 = ∅ → 𝐵 = ∅)
43uneq2d 3169 . . . 4 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∪ ∅))
5 un0 3335 . . . 4 (𝐴 ∪ ∅) = 𝐴
64, 5syl6eq 2143 . . 3 (𝐵 = ∅ → (𝐴 𝐵) = 𝐴)
7 iuneq1 3765 . . . 4 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = 𝑥 ∈ ∅ (𝐴𝑥))
8 0iun 3809 . . . 4 𝑥 ∈ ∅ (𝐴𝑥) = ∅
97, 8syl6eq 2143 . . 3 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = ∅)
106, 9eqeq12d 2109 . 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 1296  cun 3011  c0 3302   cuni 3675   ciun 3752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-uni 3676  df-iun 3754
This theorem is referenced by: (None)
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