Theorem disjxun0 32829

Description: Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.)

Hypothesis

Ref	Expression
disjxun0.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = ∅)

Assertion

Ref	Expression
disjxun0	⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem disjxun0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjxun0.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = ∅)
2		nel02 4294	. . . . 5 ⊢ (𝐶 = ∅ → ¬ 𝑦 ∈ 𝐶)
3	1, 2	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑦 ∈ 𝐶)
4	3	rmounid 32751	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5	4	albidv 1943	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6		df-disj 5073	. 2 ⊢ (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶)
7		df-disj 5073	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
8	5, 6, 7	3bitr4g 317	1 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563   ∈ wcel 2145  ∃*wrmo 3369   ∪ cun 3905  ∅c0 4288  Disj wdisj 5072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-rmo 3370  df-v 3459  df-dif 3910  df-un 3912  df-nul 4289  df-disj 5073

This theorem is referenced by:  tocyccntz  33377