Theorem disjiunb 5132

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 5020	. 2 ⊢ (𝑖 = 𝑗 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐷 𝐸)
4	3	disjor 5124	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 1539  ∀wral 3060   ∩ cin 3949  ∅c0 4332  ∪ ciun 4990  Disj wdisj 5109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rmo 3379  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967  df-nul 4333  df-iun 4992  df-disj 5110

This theorem is referenced by:  disjiund  5133  otiunsndisj  5524  s3iunsndisj  15008