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Theorem disjiunb 5063
Description: Two ways to say that a collection of index unions 𝐶(𝑖, 𝑥) for 𝑖𝐴 and 𝑥𝐵 is disjoint. (Contributed by AV, 9-Jan-2022.)
Hypotheses
Ref Expression
disjiunb.1 (𝑖 = 𝑗𝐵 = 𝐷)
disjiunb.2 (𝑖 = 𝑗𝐶 = 𝐸)
Assertion
Ref Expression
disjiunb (Disj 𝑖𝐴 𝑥𝐵 𝐶 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ ( 𝑥𝐵 𝐶 𝑥𝐷 𝐸) = ∅))
Distinct variable groups:   𝐴,𝑖,𝑗   𝐵,𝑗,𝑥   𝐶,𝑗   𝑖,𝐸   𝐷,𝑖,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑖)   𝐶(𝑥,𝑖)   𝐷(𝑗)   𝐸(𝑥,𝑗)

Proof of Theorem disjiunb
StepHypRef Expression
1 disjiunb.1 . . 3 (𝑖 = 𝑗𝐵 = 𝐷)
2 disjiunb.2 . . 3 (𝑖 = 𝑗𝐶 = 𝐸)
31, 2iuneq12d 4952 . 2 (𝑖 = 𝑗 𝑥𝐵 𝐶 = 𝑥𝐷 𝐸)
43disjor 5054 1 (Disj 𝑖𝐴 𝑥𝐵 𝐶 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ ( 𝑥𝐵 𝐶 𝑥𝐷 𝐸) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 844   = wceq 1539  wral 3064  cin 3886  c0 4256   ciun 4924  Disj wdisj 5039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rmo 3071  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-iun 4926  df-disj 5040
This theorem is referenced by:  disjiund  5064  otiunsndisj  5434  s3iunsndisj  14679
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