Theorem rmoun 32167

Description: "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)

Assertion

Ref	Expression
rmoun	⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 → (∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))

Proof of Theorem rmoun

Step	Hyp	Ref	Expression
1		mooran2 2549	. 2 ⊢ (∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
2		df-rmo 3375	. . 3 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑))
3		elun 4148	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
4	3	anbi1i 623	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑))
5		andir 1006	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	4, 5	bitri 275	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
7	6	mobii 2541	. . 3 ⊢ (∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
8	2, 7	bitri 275	. 2 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
9		df-rmo 3375	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10		df-rmo 3375	. . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11	9, 10	anbi12i 626	. 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑) ↔ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
12	1, 8, 11	3imtr4i 292	1 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 → (∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   ∈ wcel 2105  ∃*wmo 2531  ∃*wrmo 3374   ∪ cun 3946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-rmo 3375  df-v 3475  df-un 3953

This theorem is referenced by: (None)