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Theorem disjiunb 5138
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 5026 . 2 (𝑖 = 𝑗 𝑥𝐵 𝐶 = 𝑥𝐷 𝐸)
43disjor 5130 1 (Disj 𝑖𝐴 𝑥𝐵 𝐶 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ ( 𝑥𝐵 𝐶 𝑥𝐷 𝐸) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847   = wceq 1537  wral 3059  cin 3962  c0 4339   ciun 4996  Disj wdisj 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rmo 3378  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-iun 4998  df-disj 5116
This theorem is referenced by:  disjiund  5139  otiunsndisj  5530  s3iunsndisj  15004
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