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Theorem disjiunb 4794
 Description: Two ways to say that a collection of index unions 𝐶(𝑖, 𝑥) for 𝑖 ∈ 𝐴 and 𝑥 ∈ 𝐵 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 4698 . 2 (𝑖 = 𝑗 𝑥𝐵 𝐶 = 𝑥𝐷 𝐸)
43disjor 4786 1 (Disj 𝑖𝐴 𝑥𝐵 𝐶 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ ( 𝑥𝐵 𝐶 𝑥𝐷 𝐸) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   = wceq 1632  ∀wral 3050   ∩ cin 3714  ∅c0 4058  ∪ ciun 4672  Disj wdisj 4772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rmo 3058  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059  df-iun 4674  df-disj 4773 This theorem is referenced by:  disjiund  4795
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