Theorem disjxun0 33150

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 4285	. . . . 5 ⊢ (𝐶 = ∅ → ¬ 𝑦 ∈ 𝐶)
3	1, 2	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑦 ∈ 𝐶)
4	3	rmounid 33073	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5	4	albidv 1953	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6		df-disj 5071	. 2 ⊢ (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶)
7		df-disj 5071	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃*wrmo 3365   ∪ cun 3897  ∅c0 4279  Disj wdisj 5070

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

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 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-rmo 3366  df-v 3453  df-dif 3902  df-un 3904  df-nul 4280  df-disj 5071

This theorem is used by:  tocyccntz  33687