Theorem rmoun 33137

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 2581	. 2 ⊢ (∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
2		df-rmo 3365	. . 3 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑))
3		elun 4152	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
4	3	anbi1i 636	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑))
5		andir 1026	. . . . 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 2573	. . 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 3365	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10		df-rmo 3365	. . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11	9, 10	anbi12i 640	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∈ wcel 2144  ∃*wmo 2562  ∃*wrmo 3364   ∪ cun 4149

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-rmo 3365  df-v 3452  df-un 4151

This theorem is used by: (None)