Theorem rmoun 30265

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 2615	. 2 ⊢ (∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
2		df-rmo 3114	. . 3 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑))
3		elun 4076	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
4	3	anbi1i 626	. . . . 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 278	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
7	6	mobii 2606	. . 3 ⊢ (∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
8	2, 7	bitri 278	. 2 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
9		df-rmo 3114	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10		df-rmo 3114	. . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11	9, 10	anbi12i 629	. 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑) ↔ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
12	1, 8, 11	3imtr4i 295	1 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 → (∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   ∈ wcel 2111  ∃*wmo 2596  ∃*wrmo 3109   ∪ cun 3879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-mo 2598  df-clab 2777  df-cleq 2791  df-clel 2870  df-rmo 3114  df-v 3443  df-un 3886

This theorem is referenced by: (None)