Theorem disjiunb 5138

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 5026	. 2 ⊢ (𝑖 = 𝑗 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐷 𝐸)
4	3	disjor 5129	1 ⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 𝐸) = ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 846   = wceq 1542  ∀wral 3062   ∩ cin 3948  ∅c0 4323  ∪ ciun 4998  Disj wdisj 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rmo 3377  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-iun 5000  df-disj 5115

This theorem is referenced by:  disjiund  5139  otiunsndisj  5521  s3iunsndisj  14915