Theorem disjiunb 5079

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 = 𝑗 → 𝐵 = 𝐷)
disjiunb.2	⊢ (𝑖 = 𝑗 → 𝐶 = 𝐸)

Assertion

Ref	Expression
disjiunb	⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 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

Step	Hyp	Ref	Expression
1		disjiunb.1	. . 3 ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐷)
2		disjiunb.2	. . 3 ⊢ (𝑖 = 𝑗 → 𝐶 = 𝐸)
3	1, 2	iuneq12d 4969	. 2 ⊢ (𝑖 = 𝑗 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐷 𝐸)
4	3	disjor 5071	1 ⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 𝐸) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1541  ∀wral 3045   ∩ cin 3899  ∅c0 4281  ∪ ciun 4939  Disj wdisj 5056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-11 2159  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rmo 3344  df-v 3436  df-dif 3903  df-in 3907  df-ss 3917  df-nul 4282  df-iun 4941  df-disj 5057

This theorem is referenced by:  disjiund  5080  otiunsndisj  5458  s3iunsndisj  14867