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Theorem iununir 3735
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 3586 . . . . . 6 (𝐵 = ∅ → 𝐵 = ∅)
2 uni0 3604 . . . . . 6 ∅ = ∅
31, 2syl6eq 2088 . . . . 5 (𝐵 = ∅ → 𝐵 = ∅)
43uneq2d 3094 . . . 4 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∪ ∅))
5 un0 3248 . . . 4 (𝐴 ∪ ∅) = 𝐴
64, 5syl6eq 2088 . . 3 (𝐵 = ∅ → (𝐴 𝐵) = 𝐴)
7 iuneq1 3667 . . . 4 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = 𝑥 ∈ ∅ (𝐴𝑥))
8 0iun 3711 . . . 4 𝑥 ∈ ∅ (𝐴𝑥) = ∅
97, 8syl6eq 2088 . . 3 (𝐵 = ∅ → 𝑥𝐵 (𝐴𝑥) = ∅)
106, 9eqeq12d 2054 . 2 (𝐵 = ∅ → ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) ↔ 𝐴 = ∅))
1110biimpcd 148 1 ((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  cun 2912  c0 3221   cuni 3577   ciun 3654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-sn 3378  df-uni 3578  df-iun 3656
This theorem is referenced by: (None)
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