Theorem rmoun 30743

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 2556	. 2 ⊢ (∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
2		df-rmo 3071	. . 3 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑))
3		elun 4079	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
4	3	anbi1i 623	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑))
5		andir 1005	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	4, 5	bitri 274	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
7	6	mobii 2548	. . 3 ⊢ (∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
8	2, 7	bitri 274	. 2 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃*𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
9		df-rmo 3071	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10		df-rmo 3071	. . 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 291	1 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 → (∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   ∈ wcel 2108  ∃*wmo 2538  ∃*wrmo 3066   ∪ cun 3881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2817  df-rmo 3071  df-v 3424  df-un 3888

This theorem is referenced by: (None)