Theorem disjxun0 30322

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 4291	. . . . 5 ⊢ (𝐶 = ∅ → ¬ 𝑦 ∈ 𝐶)
3	1, 2	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑦 ∈ 𝐶)
4	3	rmounid 30255	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5	4	albidv 1920	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6		df-disj 5025	. 2 ⊢ (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶)
7		df-disj 5025	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
8	5, 6, 7	3bitr4g 316	1 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534   = wceq 1536   ∈ wcel 2113  ∃*wrmo 3140   ∪ cun 3927  ∅c0 4284  Disj wdisj 5024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rmo 3145  df-v 3493  df-dif 3932  df-un 3934  df-nul 4285  df-disj 5025

This theorem is referenced by:  tocyccntz  30805