Theorem disjiunb 5096

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 4983	. 2 ⊢ (𝑖 = 𝑗 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐷 𝐸)
4	3	disjor 5088	1 ⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 𝐸) = ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570  ∀wral 3080   ∩ cin 4092  ∅c0 4281  ∪ ciun 4954  Disj wdisj 5073

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rmo 3369  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-nul 4282  df-iun 4956  df-disj 5074

This theorem is used by:  disjiund  5097  otiunsndisj  5497  s3iunsndisj  15150