Theorem disjxun0 31805

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 4333	. . . . 5 ⊢ (𝐶 = ∅ → ¬ 𝑦 ∈ 𝐶)
3	1, 2	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑦 ∈ 𝐶)
4	3	rmounid 31735	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5	4	albidv 1924	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6		df-disj 5115	. 2 ⊢ (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶)
7		df-disj 5115	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
8	5, 6, 7	3bitr4g 314	1 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540   = wceq 1542   ∈ wcel 2107  ∃*wrmo 3376   ∪ cun 3947  ∅c0 4323  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-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-rmo 3377  df-v 3477  df-dif 3952  df-un 3954  df-nul 4324  df-disj 5115

This theorem is referenced by:  tocyccntz  32303