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Theorem disjiunb 5019
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 4909 . 2 (𝑖 = 𝑗 𝑥𝐵 𝐶 = 𝑥𝐷 𝐸)
43disjor 5010 1 (Disj 𝑖𝐴 𝑥𝐵 𝐶 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ ( 𝑥𝐵 𝐶 𝑥𝐷 𝐸) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wo 846   = wceq 1542  wral 3053  cin 3842  c0 4211   ciun 4881  Disj wdisj 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-mo 2540  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rmo 3061  df-v 3400  df-dif 3846  df-in 3850  df-ss 3860  df-nul 4212  df-iun 4883  df-disj 4996
This theorem is referenced by:  disjiund  5020  otiunsndisj  5377  s3iunsndisj  14419
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