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Theorem iununir 3729
 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) = x B (Ax) → (B = ∅ → A = ∅))
Distinct variable groups:   x,A   x,B

Proof of Theorem iununir
StepHypRef Expression
1 unieq 3580 . . . . . 6 (B = ∅ → B = ∅)
2 uni0 3598 . . . . . 6 ∅ = ∅
31, 2syl6eq 2085 . . . . 5 (B = ∅ → B = ∅)
43uneq2d 3091 . . . 4 (B = ∅ → (A B) = (A ∪ ∅))
5 un0 3245 . . . 4 (A ∪ ∅) = A
64, 5syl6eq 2085 . . 3 (B = ∅ → (A B) = A)
7 iuneq1 3661 . . . 4 (B = ∅ → x B (Ax) = x ∅ (Ax))
8 0iun 3705 . . . 4 x ∅ (Ax) = ∅
97, 8syl6eq 2085 . . 3 (B = ∅ → x B (Ax) = ∅)
106, 9eqeq12d 2051 . 2 (B = ∅ → ((A B) = x B (Ax) ↔ A = ∅))
1110biimpcd 148 1 ((A B) = x B (Ax) → (B = ∅ → A = ∅))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∪ cun 2909  ∅c0 3218  ∪ cuni 3571  ∪ ciun 3648 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-uni 3572  df-iun 3650 This theorem is referenced by: (None)
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