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Theorem disjiunb 5046
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 4938 . 2 (𝑖 = 𝑗 𝑥𝐵 𝐶 = 𝑥𝐷 𝐸)
43disjor 5037 1 (Disj 𝑖𝐴 𝑥𝐵 𝐶 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ ( 𝑥𝐵 𝐶 𝑥𝐷 𝐸) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wo 841   = wceq 1528  wral 3135  cin 3932  c0 4288   ciun 4910  Disj wdisj 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rmo 3143  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-iun 4912  df-disj 5023
This theorem is referenced by:  disjiund  5047  otiunsndisj  5401  s3iunsndisj  14316
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